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THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

LOS  ANGELES 


in  tftje  City  0f  ^jeitr  ^0rk 

^jblicatio:n^  number  one 

OF    THE 

ERNEST  KEMPTON  ADAMS  FUND  FOR  PHYSICAL  RESEABOH 

Established  December  17,  1904 

FIELDS  OF'  FOBCE 

A   COURSE   OF   LECTURES   IN   MATHEMATICAL   PHYSICS 
DELIVERED  DECEMBER   1   TO*  23,    1905 


BY 


VILHELM   FRIMAN   KOREN   BJERKNES 

PROFESSOR  OF  MECHANICS  AND  MATHEMATICAL  PHYSICS  IN  THE  UNIVERSITY  OF  STOCKHOLM 
LECTURER  IN  MATHEMATICAL  PHYSICS  IN  COLUMBIA  UNIVERSITY,  1905-6 


Neto  Yotk 
THE   COLUMBIA  UNIVERSITY   PRESS 

THE  MACMILLAN  COMPANY/  gents 

LONDON :    MACMILLAN  CO.,  lJ  | 
1906     .  1   ' 


TJ- 


M 


Pisess  or 
Thc  New  En*  PniNTiNQ  company 

Uf<C<9TCII,    P*. 


in  the  City  of  ^txo  ^arU 

PUBLICATION  NUMBER  ONE 

OF     IHK 

ERNEST   KEMPTON   ADAMS   FUND   FOR  PHYSICAL  RESEAROH 

Established  Deceiniber  17,  1904 

FIELDS  OF  FOECE 

A   COURSE   OF   LECTURES   IN   MATHEMATICAL   PHYSICS 
DELIVERED   DECEMBER   1    TO   23,    1905 


BY 


VILHELM    FRIMAN   KOREN    BJERKNES 

PROFESSOR  OF  MECHANICS  AND  MATHEMATICAL  PHYSICS  IN  THE  UNIVERSITY  OF  STOCKHOLM 
LECTURER  IN  MATHEMATICAL  PHYSICS  IN  COLUMBIA   UNIVERSITY,  1905-6 


THE   COLUMBIA   UNIVERSITY   PRESS 

THE  MACMILLAN   COMPANY,    Agents 

LONDON:    MACMILLAN  CO.,  Ltd. 

1906 


y 


On  the  seventeenth  day  of  December  nineteen  hundred  and  four, 
Edward  Dean  Adams,  of  New  York,  established  in  Cohimbia 
University  "  The  Ernest  Kempton  Adams  Fund  for  -Physical 
Research"  as  a  memorial  to  his  son,  Ernest  Kempton  Adams, 
who  received  the  degrees  of  Electrical  Engineering  in  1897  and 
Master  of  Arts  in  1898,  and  who  devoted  his  life  to  scientific  re- 
search. The  income  of  this  fund  is,  by  tlie  terms  of  the  deed  of 
gift,  to  be  devoted  to  the  maintenance  of  a  research  fellowship  and 
to  the  publication  and  distribution  of  the  results  of  scientific  re- 
search on  the  part  of  the  fellow.  A  generous  interpretation  of 
the  terms  of  the  deed  on  the  part  of  Mr.  Adams  and  of  the  Trus- 
tees of  the  University  has  made  it  possible  to  print  and  distribute 
the  following  lectures  as  a  publication  of  the  Ernest  Kempton 
Adams  Fund  : 


FIELDS  OF  FORCE. 


I. 

ELEMENTARY   INVESTIGATION   OF   THE   GEO- 
METRIC PROPERTIES  OF  HYDRO- 
DYNAMIC  FIELDS. 

Introductory. 

The  idea  of  electric  and  magnetic  fields  of  force  was  intro- 
duced by  Faraday  to  avoid  the  mysterious  idea  of  an  action  at 
a  distance.  After  the  victory  which  Maxwell's  theory  gained 
through  the  experiments  of  Hertz,  the  idea  of  these  fields  took 
its  place  among  the  most  fruitful  of  theoretical  physics. 

And  yet  if  we  ask,  what  is  an  electric  or  a  magnetic  field  of 
force  ?  no  one  will  be  able  to  give  a  satisfactory  answer.  We  have 
theories  relating  to  these  fields,  but  we  have  no  idea  whatever  of 
what  they  are  intrinsically,  nor  even  the  slightest  idea  of  the  path 
to  follow  in  order  to  discover  their  true  nature.  Above  all  other 
problems  which  are  related  to  fields  of  force,  and  which  occupy 
investigators  daily,  we  have  therefore  the  problem  of  fields  of 
force,  viz.,  the  problem  of  their  true  nature. 

My  lectures  will  not  give  the  solution  of  this  problem,  but  I 
should  be  happy  if  they  should  contribute  to  a  broadening  of  our 
view  of  it. 

AVhat  I  wish  to  insist  upon  especially,  is  this.  Besides  elec- 
tric and  magnetic  fields  there  exist  other  fields  which  have  strik- 
ingly analogous  properties,  and  which  have,  therefore,  exactly  the 
same  claim  to  be  called  fields  of  force.  The  investigation  of  these 
other  fields  side  by  side  with  the  electric  and  magnetic  fields  will 
be  advantageous,  I  think,  in  broadening  our  view  of  the  problem, 
especially  as  the  true  nature  of  these  other  fields  of  force  is  per- 
1  1 


2  FIELDS    OF    FORCE. 

fectly  plain  and  intelligible,  as  intelligible,  at  least,  as  anything 
can  be  in  the  limited  state  of  onr  power  of  nnderstanding. 

These  other  fields  of  force  exist  in  material  media  which  are 
in  suitable  states  of  motion.  They  are  perfectly  intelligible  in 
this  sense,  that  their  properties  can  be  deduced  from  the  principles 
of  dynamics.  For  the  special  case  when  the  material  medium  is  a 
perfect  fluid,  the  properties  of  these  fields  have  been  extensively 
explored,  and  therefore  our  main  subject  will  be  the  investigation 
of  the  properties  of  hydrodynamic  fields  of  force  and  their  com- 
parison with  electric  or  magnetic  fields. 

The  results  which  I  shall  bring  before  you  were  discovered 
originally  by  my  late  father.  Professor  C.  A.  Bjerknes,  of  Chris- 
tiania.  But  I  will  bring  into  application  here  new  methods,  which 
allow  us  to  find  the  results  with  much  greater  generality,  and  at  the 
same  time  with  greater  facility.* 

My  lectures  will  be  divided  into  two  parts.  The  first  two  lec- 
tures will  be  devoted  to  the  development  of  the  properties  of  hy- 
drodynamic fields  by  elementary  reasoning  and  experiment ;  the 
following  lectures  will  give  the  analytical  investigation  of  the  same 
subject,  based  upon  Euler's  equations  of  motion  for  a  perfect 
fluid  and  Maxwell's  equations  for  the  electromagnetic  field. 

1.  Field-x'edovs.  —  The  electric  field  may  be  described  in  the 
simplest  case  by  either  of  two  vectors,  the  electric  jinx  (or  dis- 
placement), or  the  electric  field  intensity  (or  force).  In  the  same 
way  the  magnetic  field  may  be  described  by  the  magnetic  flux  (or 
induction),  or  the  magnetic ^le/fZ  intensity  (or  force).  The  flux  and 
the  field  intensity  will  differ  from  each  other  only  by  a  constant 
factor,  the  electric  or  the  magnetic  inrJiirtivity  of  the  medium  which 
supports  the  field,  the  flux  being  always  the  product  of  the  field 
intensity  into  the  inductivity. 

*  For  the  historical  development  of  C  A.  Bjerknes'  ideas  compare  z'.  Bjerknes: 
C.  A.  Bjerknes,  d'ediiehtnissrefle  gehalten  in  der  Geselhchnfl  cler  Wissenschuften  zu 
Christiania  dm  April,  1903.  German  translation,  Leipzig,  1903.  For  the  de- 
velopment of  the  theory  according  to  C.  A.  Bjerknes'  methods,  and  for  more 
complete  description  of  instruments  and  experiments,  see  v.  Bjerknes,  Vorlesungen 
Ober  hxjrhofhjnnmi»che  Fernkriifte  nach  C  A.  Bjerknes'  Theorie.  Vols.  I  and  II, 
Leipzig,  1900-02. 


INVESTIGATION    OF    GEOMETRIC    PROPERTIES.  3 

Ou  the  otlier  hand,  the  fiehl  of  motion  in  any  moving  liquid 
may  also  be  described  by  either  of  two  vectors,  which  are 
related  to  each  other  in  the  same  way  as  the  flux  and  the  field 
intensity  of  electric  or  magnetic  fields.  The  first  of  these 
vectors  is  the  velocity,  and  the  other  the  j^roduct  of  the  velocity 
into  the  density.  As  to  its  dynamical  significance,  this  vector  is 
the  momentum  per  unit  volume  or  the  specific  momentum  in  the 
moving  fluid. 

In  hydrodynamics  we  thus  meet  with  two  vectors  which  are 
connected  in  a  similar  way  as  the  flux  and  the  field  intensity  in  the 
electric  or  magnetic  field.  This  parallelism  at  once  invites  a  com- 
parison. There  is  only  the  question  as  to  how  the  different  vec- 
tors should  be  paired,  and  this  can  be  answered  only  by  a  closer 
analysis  of  their  properties.  This  brings  us  to  the  question  of  our 
knowledge  of  the  fields. 

2.  Geometric  and  Dynamic  Properties  of  tlie  Fields.  — The  extent 
of  our  knowledge  of  the  diiferent  kinds  of  fields  differs  greatly.  All 
the  properties  of  the  hydrodynamic  fields  follow  directly  from  the 
most  trustworthy  laws  of  nature,  that  is  from  the  principle  of  the 
conservation  of  the  mass,  and  from  the  principles  of  dynamics. 
With  reference  to  electric  or  magnetic  fields,  on  the  contrary,  we 
have  only  formal  theories.  First,  we  have  an  extensively  devel- 
oped geometric  theory  of  the  distribution  in  space  of  the  vectors 
which  describe  the  field.  And  then,  in  a  more  or  less  superficial 
connection  to  this  geometric  theory,  we  have  a  very  much  less 
developed  theory  of  the  dynamic  j^f'ojjerties  of  the  fields. 

Taking  the  facts  as  they  lie  before  us,  we  shall  be  obliged, 
therefore,  to  give  to  our  theory  a  dualistic  form,  comparing 
separately  the  geometric  and  the  dynamic  properties  of  the  two 
kinds  of  fields.  It  may  be  reserved  for  the  future  to  penetrate  to 
the  central  point,  where  the  geometry  and  the  dynamics  of  the 
question  are  perfectly  united,  and  thus  make  the  comparison  of 
the  two  kinds  of  fields  perfectly  easy. 

In  this  lecture  we  will  consider  the  geometric  properties  of  the 
fields. 


4  FIELDS    OF    FORCE. 

3.  Properties  of  the  Field  Vectors  at  a  Surface  of  Separation. — 
A  characteristic  geometrical  property  of  the  vectors  at  a  surface  of 
separation  of  two  media  shows  at  once  liow  the  fluxes  and  field 
intensities  should  be  paired  with  the  hydrodynamic  vectors.  As 
is  well  known,  at  a  surfiice  of  separation  the  normal  component 
of  the  flux  is  always  continuous,  while  the  normal  component  of 
the  field  intensity  is  necessarily  discontinuous,  if  the  inductivity 
suddenly  changes  at  the  surface.  On  the  other  hand,  at  any  sur- 
face of  separation  in  a  moving  liquid  the  normal  component  of  the 
velocity  is  continuous,  corresponding  to  the  normal  component  of 
the  flux.  Otherwise  we  should  have  at  the  surface  either  creation 
or  annihilation  of  matter,  or  a  break  in  the  continuity,  both  of 
which  we  consider  excluded.  From  the  continuity  of  the  normal 
component  of  the  velocity  follows  the  discontinuity  of  the  specific 
momentum  for  tiie  case  where  the  density  of  the  fluid  suddenly 
changes  at  the  surface.  This  vector,  therefore,  has  a  discontinuity 
similar  to  that  of  the  field  intensity,  and  it  follows  at  once  that  the 
correspondence  of  the  vectors  is  possible  if  the  velocity  correspond 
to  the  flux  and  the  specific.niomentum  to  the  field  intensity.  Fur- 
ther, as  the  flux  is  the  product  of  the  field  intensity  into  the  induc- 
tivity, it  also  follows  that  not  the  density,  but  the  reciprocal  of  the 
density,  or  the  specific  volume,  corresponds  to  the  inductivity. 

Just  as  the  density  gives  the  measure  of  the  inert  resistance  of 
the  matter  to  the  motion,  the  specific  volume  gives  the  measure  of 
the  readiness  of  the  matter  to  take  inotion.  The  specific  volume 
may  therefore  also  be  termed  the  m.obilitij  of  the  fluid.  We  thus 
get  the  correspondence  : 

velocity flux, 

sj)ecific  momentum field  intensity, 

mobility inductivity. 

We  have  now  to  examine  more  closely  the  content  of  this 
correspondence. 

4.  Charged  Particle — Expandiiu/  or  Contracting  Particle. — 
TiCt  us  start  with  the  simplest  object  met  with  in  the  first  investi- 


INVESTIGATION    OF    GEOMETRIC    TROPERTIES.  5 

gations  of  electricity,  namely,  with  an  electrically  charged  particle. 
In  the  field  belonging  to  this  particle  the  vectors  are  directed 
radially  outwards  if  the  particle  has  a  positive,  and  radially  in- 
wards if  it  has  a  negative  charge,  and  their  intensity  decreases  as 
the  inverse  square  of  the  distance. 

It  is  seen  at  once  that  an  expanding  particle  which  is  contained 
in  an  incompressible  fluid,  such  as  water,  will  produce  a  field  of 
exactly  the  same  geometrical  nature  as  the  field  belonging  to  the 
positively  charged  particle.  It  will  produce  a  radial  current 
directed  outwards,  in  which,  as  a  consequence  of  the  incompressi- 
bility,  the  velocity,  and  therefore  the  specific  momentum  will  de- 
crease as  the  inverse  square  of  the  distance.  In  like  manner  a  con- 
tracting particle  will  be  surrounded  by  a  current  directed  radially 
inwards,  and  will  thus  correspond  to  a  negatively  charged  particle 
(see  Fig.  4,  a  and  b,  below). 

This  comparison  of  a  radial  electric  and  a  radial  hydrodynamic 
field  has  one  difficulty,  however.  The  idea  of  an  always  expand- 
ing, or  of  an  always  contracting  particle,  is  impossible.  Therefore 
to  make  the  comparison  possible,  not  only  for  a  moment,  but  for 
any  length  of  time,  we  are  obliged  to  consider  a  motion  of  peri- 
odic expansions  and  contractions,  or  a  pulsating  motion.  In  this 
case  there  wnll  be  no  absolute  diiference  between  a  positive  and  a 
negative  pulsating  body.  But  two  pulsating  bodies  may  be  in 
exactly  the  same  mutual  relation  to  each  other  as  an  expanding 
and  a  contracting  body.  For  if  they  are  pulsating  in  oppo- 
site phase,  the  one  will  always  be  expanding  while  the  other  is 
contracting,  and  vice  versa.  We  can  then  distinguish  these  two 
pulsating  bodies  from  each  other  by  opposite  signs,  just  as  w^e 
do  two  oppositely  charged  particles,  and  we  can  represent  the 
mean  state  of  motion  in  the  surrounding  radial  field  by  a  vector 
directed  outwards  from  the  pulsating  particle  which  we  call,  by 
convention,  positive,  and  inwards  to  the  pulsating  particle  which 
we  call  negative. 

5.  Complex  Fields.  —  If  we  consider  two  charged  particles 
whose  dimensions  are  sufficiently  small  in  comparison  to  the  dis- 


FIELDS    OF    FORCE. 


tance  between  them,  a  field  is  produced  which  is  the  simple  result 
of  the  superposition  of  the  two  radial  fields.  The  lines  of  force  of 
the  complex  fields  may  be  found  by  the  well-known  constructions 
of  superposition.  The  results  are  the  well  known  curves  running 
from  one  charged  particle  to  the  other  for  the  case  of  opposite 
charges,  and  the  diverging  curves  with  a  neutral  point  between 
the  two  charged  particles  for  the  case  of  charges  of  the  same  sign 
(see  Figs.  5  and  6  below). 

In  exactly  the  same  way,  if  we  consider  two  particles  which 
have  a  motion  of  expansion  or  contraction  and  which  are  suffi- 
ciently small  in  comparison  to  the  distance  between  them,  the 
radial  currents  produced  by  each  will  simply  be  superposed,  and 
the  current  lines,  by  which  the  complex  field  may  be  represented, 
can  be  found  by  exactly  the  same  construction  as  in  the  case  of  the 
corresponding  electric  fields.  And  this  result  may  be  transferred 
at  once  to  the  case  of  vibratory  motion  ;  particles  pulsating  in  the 
same  phase,  expanding  simultaneously  and  contracting  simul- 
taneously, will  give  a  field  corresponding  geometrically  to  that 
produced  by  particles  carrying  charges  of  the  same  sign,  and 
oppositely  pulsating  particles  will  produce  a  field  corresponding 
geometrically  to  that  produced  by  particles  which  carry  opposite 
charges. 

Just  as  we  combine  the  fields  of  two  charged  particles,  we  can 
combine  the  fields  of  any  inunber  of  charged  particles,  and  to  a  field 
of  any  complexity  obtained  in  this  way  we  can  construct  a  corre- 
sponding hydrodyuamic  field,  obtained  by  the  combination  of  the 
fields  of  the  corresponding  system  of  expanding  and  contracting 
particles,  or  of  pulsating  particles  for  the  case  of  vibratory  motion. 
An  extensive  geometric  analogy  between  hydrodyuamic  and  elec- 
trostatic fields  is  thus  found. 

(i.  hitv'nmralhi  Polarized  Bodies!.  0-'<cil/afinr/  Bodie>!.  ^  What 
we  have  said  of  electrified  particles  and  tiie  electric  fields  produced 
by  them  may  l)e  repeated  for  magnetic  poles  and  the  correspond- 
ing magnetic  fields.  But  now  tiie  reservation  must  be  made,  that 
magnetic  ])oles  are  in  reality  mere  fictions.     For  a  distribution  of 


INVESTIGATION    OF    GEOMETRIC    PROPERTIES.  7 

magnetic  poles  we  can,  however,  substitute  a  state  of  intrinsic 
polarization,  Avhich  may  be  considered  as  the  real  origin  of  the 
mao-netic  field.  Such  states  of  intrinsic  polarization  are  also  met 
with  in  electricity.  Thus  the  pyro-electric  crystal  seems  to  give  a 
perfect  electric  analogy  to  the  permanent  magnet. 

Let  us  now  for  the  system  of  magnetic  poles,  by  which  a  mag- 
net can  be  represented  symbolically,  substitute  the  corresponding 
system  of  expanding  and  contracting  particles.  In  the  region  of 
the  fluid  which  corresponds  to  the  magnet  the  total  sum  of  ex- 
pansions and  contractions  will  be  zero.  But  the  field  produced 
in  the  exterior  space  by  these  expansions  and  contractions  may 
also  be  produced  by  quite  another  interior  motion,  involving  no 
expansion  or  contraction  at  all.  For  consider  a  closed  surface 
consisting  of  fluid  particles,  and  surrounding  the  region  of  the 
fluid  which  corresponds  to  the  magnet.  This  material  surface  has 
a  certain  motion  ;  it  will  advance  on  that  side  where  the  expand- 
ing particles  are  situated,  and  recede  on  that  side  where  the  con- 
tracting particles  are  situated.  The  result  is  a  motion  of  the  sur- 
face as  a  whole,  directed  from  the  regions  of  contraction  towards 
the  regions  of  expansion.  And,  as  the  sum  of  the  expansions  and 
the  contractions  is  zero,  the  volume  within  the  surface  will  remain 
unchanged  during  this  motion. 

Now  the  motion  produced  outside  the  surface  will  be  entirely 
independent  of  what  goes  on  within  it,  provided  only  that  the 
motion  of  the  surface  itself  remains  unchanged.  We  can  there- 
fore do  away  with  the  expansions  and  contractions,  and  suppose 
the  volume  within  the  surfiice  filled  with  an  incompressible  fluid, 
subject  to  the  action  of  forces  which  give  these  fluid  masses  a 
motion  consistent  with  the  required  motion  of  the  surface. 

We  have  thus  arrived  at  the  following  result  :  a  motion  of  in- 
compressible fluid  masses,  produced  by  suitable  forces,  can  be 
found,  which  will  set  up  an  exterior  field  similar  to  that  set  up  by 
a  system  of  expanding  and  contracting  particles,  provided  that  the 
sum  of  the  expansions  and  contractions  is  zero.  And  this  equivalence 
corresponds  exactly  to  the  equivalence  between  the  representation 


8  FIELDS    OF    FORCE. 

of  a  magnet  by  a  distribution  of  poles,  and  by  a  state  of  intrinsic 
polarization.  The  hydrodynamic  model  of  a  body  in  a  state  of 
intrinsic  polarization  is,  therefore,  a  body  consisting  of  incompressi- 
ble fluid  masses,  moved  through  the  surrounding  fluid  by  suitable 
exterior  forces  (see  Fig.  8  below). 

We  have  considered' here,  for  simplicity,  only  the  instantaneous 
state  of  motion.  In  the  case  of  periodic  motion  we  get  an  equiv- 
alence between  a  system  of  oppositely  pulsating  particles  and  a 
fluid  body  which  takes  forced  oscillations  under  the  influence  of 
suitable  exterior  forces. 

7.  Fields  in  Hcferoc/eneous  Media.  —  The  results  already  de- 
veloped depend,  essentially,  upon  the  supposition  that  the  fluid 
surrounding  the  moving  bodies  is  homogeneous  and  incompres- 
sible. The  case  when  it  is  heterogeneous  must  be  examined 
separately.  That  the  heterogeneity  has  an  influence  upon  the 
geometric  configuration  of  the  field,  is  obvious.  For  only  when 
the  fluid  is  perfectly  homogeneous  will  there  exist  that  perfect 
symmetry  in  the  space  surrounding  an  expanding  particle,  which 
entitles  us  to  conclude  that  a  perfectly  symmetrical  radial  current 
will  arise.  But  if  on  one  side  of  the  expanding  particle  there  ex- 
ists a  region  where  the  fluid  has  a  different  density,  the  symmetry  is 
lost,  and  it  is  to  be  expected  that  the  configuration  of  the  field 
will  be  influenced  by  tliis  fact.  On  the  other  hand,  as  is  well 
known,  any  heterogeneity  of  the  dielectric  has  a  marked  influence 
upon  the  geometric  configuration  of  the  electric  field,  giving  rise 
to  the  ])henomena  of  electrification  by  influence. 

Now,  will  the  influence  of  the  heterogeneity  in  the  two  cases  be 
of  similar  nature?  To  examine  this  question  we  shall  have  to 
develop  a  very  simple  principle  relating  to  the  dynamics  of  fluids, 
our  considerations  above  having  been  based  only  on  the  principle 
of  the  conservation  of  mass. 

8.  Principle  of  Kinetic  Buoyancy. — Consider  a  cylinder,  with  axis 
vertical,  containing  a  body  and,  apart  from  the  body,  completely 
filled  with  water.  The  condition  of  equilibrium  will  depend  upon 
the  l)uoyancy,  according  to  the  Archimedian  principle.     If  the  body 


INVESTIGATION    OF    GEOMETRIC    PROPERTIES.  9 

has  exactly  the  density  of  the  water,  the  buoyancy  will  balance  the 
weight  of  the  body,  and  it  will  remain  in  equilibrium  in  any  posi- 
tion. If  it  be  lighter,  its  buoyancy  will  be  greater  than  its  weight, 
and  it  will  tend  to  move  upwards.  If  it  be  heavier,  its  buoyancy 
will  be  less  than  its  weight,  and  it  will  tend  to  move  downwards. 
Thus,  if  we  have  three  cylinders,  each  containing  one  of  three  such 
bodies,  the  light  body  will  rise  to  the  top,  the  heavy  body  will 
sink  to  the  bottom,  and  the  body  of  the  same  density  as  the  water 
will  remain  in  any  position. 

This  static  buoyancy  depends  upon  the  action  of  gravity. 
But  there  exists  a  corresponding  dynamic  buoyancy,  which  is 
easily  observed  as  follows:  To  do  away  with  the  influence  of 
gravity,  lay  the  cylinders  with  their  axes  horizontal,  and  let  the 
bodies  be  in  the  middle  of  the  cylinders.  Then  give  each  cyl- 
inder a  blow,  so  that  they  move  suddenly  five  or  ten  centimeters 
in  the  direction  of  their  axes.  The  following  results  will  then  be 
observed  : 

1.  The  body  which  is  lighter  than  the  water  has  moved 
towards  the  front  end  of  its  cylinder,  and  thus  has  had  a  motion 
through  the  water  in  the  direction  of  the  motion  of  the  water. 

2.  The  body  which  has  the  same  density  as  the  water  has 
moved  exactly  the  same  distance  as  the  water,  and  thus  retained 
its  position  relative  to  the  water. 

3.  The  body  which  is  heavier  than  the  water  has  moved  a 
shorter  distance  than  its  cylinder,  and  thus  has  had  a  motion 
through  the  water  against  the  direction  of  motion  of  the  water. 

If  we  give  the  cylinders  a  series  of  blows,  the  light  body  will 
advance  through  the  water  nntil  it  stops  against  the  front  end.  The 
body  of  the  same  density  as  the  water  will  retain  its  place,  and  the 
heavy  body  will  move  backwards  relatively  to  the  cylinder,  until 
it  stops  against  the  end.  The  effect  is  strikingly  analogous  to 
the  effect  of  statical  buoyancy  for  the  case  of  the  cylinders  with 
vertical  axes,  and  this  analogy  exists  even  in  the  quantitative 
laws  of  the  phenomenon. 

These  quantitative  laws  are  complicated  in  case  the  bodies  are 
2 


10  FIELDS    OF    FORCE. 

free  to  move  through  the  water,  but  exceedingly  simple  when  they 
are  held  in  an  invariable  position  relative  to  the  water  by  the 
application  of  suitable  exterior  forces. 

This  exterior  force  is  nil  in  the  case  when  the  body  has  the 
same  density  as  the  water.  The  body  then  follows  the  motion  of 
the  surrounding  water  masses,  subject  only  to  the  force  resulting 
from  the  pressure  exerted  by  them.  The  motion  of  the  body  is 
subject  to  the  fundamental  law  of  dynamics, 

force  =  mass  x  acceleration. 

As  the  l)ody  has  both  the  acceleration  and  the  density  of  the 
surrounding  water  masses,  the  force  is  equal  to  the  product  of  the 
acceleration  into  the  mass  of  the  water  displaced  by  the  body. 
And  this  law  evidently  will  be  true  even  for  the  heavy  or  the 
light  body,  provided  only  that  they  are  held  by  suitable  forces  at 
rest  relatively  to  the  moving  water.  For,  the  state  of  motion  out- 
side the  body  is  then  unchanged,  and  the  pressure  exerted  by  the 
water  against  any  surface  does  not  at  all  depend  upon  the  condi- 
tions within  the  surface.-  Thus  we  find  this  general  result,  which 
is  perfectly  analogous  to  the  Archimedian  law  : 

Any  hoiJy  which  'participates  in  the  translatory  motion  of  a  fluid 
mass  is  subject  to  a  kinetic  buoyancy  equal  to  the  product  of  the 
acceleration  of  the  translatory  motion  multiplied  by  the  mass  of  water 
displaced  by  the  body. 

This  law  obviously  gives  also  the  value  of  the  exterior  force 
which  must  be  applied  in  order  to  make  the  body  follow  exactly 
the  motion  of  the  fluid,  just  as  the  Archimedian  law  gives  the 
force  which  is  necessary  to  prevent  a  body  from  rising  or  sinking. 
This  force  is  nil,  if  the  body  has  the  same  density  as  the  water,  it 
is  directed  against  the  direction  of  the  acceleration,  if  the  body  is 
lighter,  and  in  the  direction  of  the  acceleration,  if  the  body  is 
heavier.  And,  if  no  such  force  act,  we  get  the  result,  illustrated 
by  the  experiment,  that  the  light  body  moves  faster  than  the 
water  and  tlic  iieavy  body  shnver,  and  thus,  relatively,  against  the 
water. 


INVESTIGATION    OF    GEOMETRIC    PROPERTIES.  H 

9.  Influence  of  Heterogeneitie.^  in  the  Electric  or  Magnetic  and 
in  the  Analogous  Hydvodynamic  Field.  — From  the  principle  of 
kinetic  buoyancy  we  thus  find  the  obvious  law,  that,  in  a  hetero- 
geneous fluid,  masses  of  greater  mobility  take  greater  velocities. 
The  mobility  therefore  influences  the  distribution  of  velocity,  just 
as  the  inductivity  influences  the  distribution  of  the  flux  in  the 
electric,  or  magnetic  field.  For  at  places  of  greater  inductivity  we 
have  greater  electric,  or  magnetic  flux. 

To  consider  a  simple  example,  let  us  place  in  a  bottle  filled  with 
water  a  light  sphere,  a  hollow  celluloid  bdl,  for  instance,  attached 
below  with  a  fine  string.  And  in  another  bottle  let  us  suspend  in  a 
similar  manner  a  lead  ball.  If  we  shake  the  bottles,  the  celluloid 
ball  will  take  very  lively  oscillations,  much  greater  than  those  of 
the  water,  while  the  lead  ball  will  remain  almost  at  rest.  With 
respect  to  their  induced  oscillations,  they  behave,  then,  exactly  as 
magnetic  or  diamagnetic  bodies  behave  with  respect  to  the  induced 
magnetization  when  they  are  brought  into  a  magnetic  field  ;  the 
light  body  takes  greater  oscillations  than  the  water,  just  as  the 
magnetic  body  takes  greater  magnetization  than  the  surrounding 
medium.  The  heavy  body,  on  the  other  hand,  takes  smaller  oscil- 
lations than  the  water,  just  as  the  diamagnetic  body  takes  smaller 
magnetization  than  the  surrounding  medium.  And  thus  relatively, 
the  heavy  body  has  oscillations  opposite  to  those  of  the  water,  just 
as  the  diamagnetic  has  a  relative  polarity  opposite  to  that  of  the 
surrounding  medium. 

10.  Bef faction  of  the  Lines  of  Flow.  —  The  influence  which  the 
greater  velocity  of  the  masses  of  greater  mobility  has  upon  the 
course  of  the  tubes  of  flow  is  obvious.  At  places  of  greater 
velocity  the  tubes  of  flow  narrow,  and  at  places  of  smaller  velocity 
widen.  They  will  thus  be  narrow  at  places  of  great,  and  wide  at 
places  of  small  mobility,  just  as  the  tubes  of  flux  in  the  electric  or 
magnetic  field  are  narrow  at  places  of  great,  and  wide  at  places  of 
small  inductivity.  If  we  limit  ourselves  to  the  consideration  ot 
the  most  practical  case,  when  the  values  of  the  mobility  or  of  the 
inductivity  change  abruptly  at  certain  surfaces,  we  can  easily  prove 


12  FIELDS   OF    FORCE. 

that  tlie  influence  of  the  hete^'ogeneity  in  the  two  kinds  of  fields 
corresponds  not  only  qualitativity  but  quantitatively. 

AVe  suppose  that  the  bodies  which  have  other  density  than  the 
surrounding  fluid  are  themselves  fluid.  It  is  only  in  experiments 
that,  for  practical  reasons,  we  must  always  use  rigid  bodies.  At  the 
surface  of  separation  between  the  surrounding  fluid  and  the  fluid 
body  the  pressure  must  have  the  same  value  on  both  sides  of  the 
surface.  This  is  an  immediate  consequence  of  the  principle  of  equal 
action  and  reaction.  From  the  equality  of  the  pressure  on  both 
sides  of  the  surface  it  follows,  that  the  rate  of  decrease  of  the  pres- 
sure in  direction  tangential  to  the  surface  is  also  equal  at  adjacent 
points  on  each  side  of  the  surface.  But  this  rate  of  decrease  is  the 
gradient,  or  the  force  per  unit  volume,  in  the  moving  fluid.  And, 
as  the  acceleration  produced  by  the  force  per  unit  v^olume  is  in- 
versely proportional  to  the  density,  we  find  that  the  tangential  ac- 
celeration on  tlie  two  sides  of  the  surface  of  separation  will  be 
inversely  proportional  to  the  density.  Or,  what  is  the  same  thing, 
tlie  product  of  tlie  tnngeiitial  acceleration  into  the  density  will  have  the 
same  value  on  both  sides  of  the  surface. 

From  this  result  there  can  not  at  once  be  drawn  a  general  con- 
clusion on  the  relation  of  the  tangential  components  of  the  velocity, 
or  of  the  specific  momentum.  For  two  adjacent  particles,  which 
are  accelerated  according  to  this  law,  will  at  the  next  moment  no 
longer  be  adjacent.  If,  however,  the  motion  be  periodic,  so  that 
every  particle  has  an  invariable  mean  position,  then  adjacent  par- 
ticles will  remain  adjacent  particles,  and  from  the  equality  of  the 
tangential  components  of  the  products  of  the  accelerations  into  the 
densities  at  once  follows  the  equality  of  the  tangential  components 
of  the  products  of  the  velocities  into  the  densities.     Thus, 

//(  the  case  of  vibratory  motion  the  specific  momentum  lias  con- 
tinuous tanf/ential  components  at  the  surface  of  separation  of  two 
media  of  different  mobility. 

The  law  for  the  specific  momentum  is  thus  exactly  the  same 
as  for  the  electric  or  magnetic  field  intensities,  which  have  con- 
tinuous tangential  components  at  the  surface  of  separation  of  two 


INVESTIGATION    OF    GEOMETRIC    PROPERTIES.  13 

media  of  different  inductivity.  As  we  have  already  found  (3), 
the  law  for  the  velocity  is  the  same  as  for  the  electric  or  the  mag- 
netic flux.  We  see  then,  that  the  conditions  fulfilled  at  a  surface 
of  separation  by  the  hydrodynamic  vectors  on  the  one  hand,  and 
by  the  electric  or  magnetic  vectors  on  the  other,  are  identically 
the  same.  The  lines  of  flow  and  the  lines  of  flux  will  show 
exactly  the  same  peculiarity  in  passing  a  surface  of  separation. 
And,  as  is  shown  in  all  treatises  on  electricity,  this  peculiarity 
consists  in  a  refraction  of  the  lines  so  that  the  tangents  of  the 
angles  of  incidence  and  refraction  are  in  the  same  ratio  as  the  induc- 
tivities  on  the  two  sides  of  the  surface.  In  the  hydrodynamic 
case  these  tangents  will  be  in  the  same  ratio  as  the  mobilities  on 
the  two  sides  of  the  surface.  This  refraction  gives  to  the  tubes  of 
flow  or  of  flux  the  sudden  change  of  section  which  corresponds  to 
the  increase  or  decrease  of  the  velocity  or  of  the  flux  in  passing 
from  one  medium  into  the  other. 

This  refraction  of  the  hydrodynamic  lines  of  flow  according  to 
the  same  law  as  that  of  the  refraction  of  the  electric  or  magnetic 
lines  of  force  is  a  phenomenon  met  with  daily  in  the  motion  of  super- 
imposed liquids  of  different  specific  weights.  If  I  suddenly  move  a 
glass  partly  filled  with  mercury  and  partly  with  water,  the  mercury 
rises  along  the  rear  wall  of  the  glass,  while  the  water  sinks  in  front. 
During  the  first  instant  of  the  motion,  before  we  get  the  oscillations 
due  to  gravity,  the  law  of  the  refraction  of  the  tubes  of  flow  is  ful- 
filled at  the  surface  of  separation.  Whatever  be  the  course  of  the 
tubes  of  flow  at  a  distance  from  the  surface,  at  the  surface  they 
will  be  refracted  so  that  the  tangents  of  the  angles  of  incidence 
and  of  refraction  are  in  the  ratio  of  the  mobilities  of  the  mercury 
and  of  the  water,  or  in  the  inverse  ratio  of  their  densities,  1  :  13. 

We  get  the  same  law  of  refraction  at  the  surface  of  separation 
of  water  and  air,  the  tangents  of  the  angles  being  then  in  the 
ratio,  1  :  700.  The  accident  of  daily  occurrence,  in  which  a  glass 
of  water  flows  over  as  the  result  of  sudden  motion,  is  thus  the  conse- 
quence of  a  law  strictly  analogous  to  that  of  the  refraction  of  the 
electric  or  magnetic  lines  of  force. 


14  FIELDS    OF    FOECE. 

11.  Experimental  Verijieations.  —  We  have  been  able  from 
kinematic  and  dynamic  principles  of  the  simplest  nature  to  show 
the  existence  of  an  extended  analogy  in  the  geometric  properties  of 
the  electric  or  magnetic,  and  hydrodyuamic  lields.  The  dynamic 
principles  which  form  the  basis  of  this  analogy  we  have  illus- 
trated by  experiments  of  the  simplest  possible  nature.  But  even 
though  we  have  perfect  faith  in  the  truth  of  the  results,  it  is 
desirable  to  see  direct  verifications  of  them.  Some  experiments 
have  been  made  towards  finding  verifications,  but  not  as  many, 
however,  as  might  have  been  desirable. 

These  experiments  were  made  with  water  motions  of  vibra- 
tory nature,  produced  by  pulsating  or  oscillating  bodies,  using 
instruments  constructed  mainly  for  the  investigation  of  the  dyna- 
mic properties  of  the  field,  which  will  be  the  subject  of  the  next 
lecture.  Such  pulsations  and  oscillations  can  easily  be  produced 
by  a  pneumatic  arrangement  involving  a  generator  which  pro- 
duces an  alternating  current  of  air. 

12.  The  Generator. — A  generator  of  this  kind  consists  of  two 
small  air  pumps  of  the  'simplest  possible  construction,  without 
valves.  To  avoid  metal  work  we  can  simply  use  drums, 
covered  with  rubber  membranes,  which  are  alternately  pressed  in 
and  drawn  out.  These  pumps  should  be  arranged  so  that  they 
can  work  in  either  the  same  or  in  opposite  phase,  and  so  that  the 
am])litudes  of  the  strokes  of  each  pump  can  be  varied  indepen- 
dently of  the  other.  For  convenience,  it  should  be  possible  to 
reverse  the  ])hase  and  vary  the  amplitudes  without  interrupting 
the  motion  of  the  generator. 

In  Fig.  1  is  shown  a  generator,  arranged  to  fulfill  these  con- 
ditions. In  a  wooden  base  are  fixed  two  vertical  steel  or  brass 
sjjrings,  s,  which  are  joined  by  the  horizontal  connecting-rod,  6. 
The  upper  ends  of  tiiese  springs  are  connected  by  the  piston-rods, 
a,  to  the  pistons  of  the  air-pumps,  which  are  supported  on  a 
wooden  frame  in  such  a  way  that  each  is  free  to  turn  about  a 
iiorizontal  axis,  c,  passing  through  the  top  of  the  corresponding 
spring  perpendicular  to  the  piston-rod.     Thus  either  pump  can  be 


INVESTIGATION    OF    GEOMETKIC    PEOPEimES. 


15 


revolv'ed  through  180°,  or  tlirough  a  smaller  angle,  without 
stopping  the  pumps.  The  amplitude  of  the  strokes  in  any  posi- 
tion is  proportional  to  the  cosine  of  this  angle,  since  the  compo- 
nent of  the  motion  of  the  top  of  the  spring  along  the  axis  of  the 
cylinder  is  proportional  to  this  cosine.  At  90°  the  amplitude  is  0, 
and  the  phase  changes,  so  that  by  a  simple  rotation  we  are  able  to 
reverse  the  phase,  or  vary  the  amplitude  of  either,  or  both  pumps- 
The  generator  may  be  driven  by  a  motor  of  suitable  nature, 
attached  to  the  frame.  As  shown  in  the  figure,  we  may  use  a 
fly-wheel,  d,  carrying  a  crank  which  drives  the  springs,  using  an 


Fig.  1. 

electric  motor,  or  any  other  suitable  source,  for  motive  power. 
The  use  of  the  crank  has  the  advantage  that  the  amplitudes  of  the 
oscillations  of  the  springs  are  invariable  and  independent  of  the 
resistance  to  the  motion.  It  should  be  noted  here,  that,  with  the 
crank,  the  springs  may  be  used  simply  as  rigid  levers,  by  loosen- 
ing the  screws,  m,  which  hold  them  in  the  base.  The  springs  are 
then  free  to  turn  about  a  pivot  just  below  the  screws. 

A  hydraulic  motor  might  also  be  used  to  drive  the  generator. 
Two  coaxial  brass  cylinders,  open  at  the  same  end,  are  so  ar- 
ranged that  the  inner  projects  slightly  beyond  the  outer.  A  rubber 
membrane  is  stretched  over   the  open  ends  of  the  two  tubes,  so 


16 


FIELDS    OF    FORCE. 


that  water  admitted  to  the  outer  cylinder  cannot  pass  into  the 
inner  cylinder  without  pressing  out  the  membrane.  Under  suita- 
ble circumstances,  tliis  produces  a   vibration  of  the  membrane, 


Fig.  2. 


Fkj.  3. 


which  can  be  communicated  to  the  pumps  by  the  connecting-rods. 
The  period  will  d('j)end  upon  the  tension  of  the  membrane,  the 


INVESTIGATION    OF    GEOMETRIC    PROPERTIES.  17 

stiffness  of  the  springs,  and  the  length  and  section  of  the  dis- 
charge-pipe. An  electromagnetic  vibrator  is  often  convenient  for 
driving  the  generator. 

13.  Puhalor.  Oscilhdor.  —  For  a  pulsating  body  we  may  use 
an  india-rubber  balloon  attached  to  one  end  of  a  metal  tube,  the 
other  end  of  which  is  connected  by  a  rubber  tube  with  one  of 
the  pumps  of  the  generator.  As  the  balloon  often  takes  irregular 
forms  and  motions,  it  is  usually  more  convenient  to  let  the  tube 
end  in  a  drum,  which  is  covered  on  each  side  with  a  rubber 
membrane.     A  diagram  is  given  in  Fig.  2. 

A  convenient  form  of  oscillator  is  shown  in  Fig.  3.  The  oscil- 
lating body  is  a  hollow  celluloid  sphere,  «,  made  in  two  halves, 
and  attached  to  a  tube  of  the  same  material,  b,  which  reaches 
above  the  surface  of  the  water.  A  metal  tube,  c,  connected  with 
one  pump  of  the  generator,  supports  the  sphere  by  pivots  at  h,  and 
terminates  in  a  heavy  drum,  d,  in  the  center  of  the  sphere.  The 
rubber  membrane,  e,  is  connected  with  one  side  of  the  sphere  by  a 
rod,  f,  so  that. the  alternating  air  current  produces  oscillations  in 
the  sphere  and  in  the  drum.  The  sphere  is  made  as  light  as  pos- 
sible and  the  drum  heavy,  so  that,  while  the  former  takes  large 
oscillations,  the  latter  will  take  very  small  oscillations  because  of 
its  greater  mass.  For  convenience  in  recognizing  the  axis  of 
oscillation  the  two  halves  of  the  sphere  may  be  painted  in  differ- 
ent colors,  so  that,  at  any  moment,  the  advancing  hemisphere  is 
one  color  and  the  receding  hemisphere  another.  Thus,  two  oscil- 
lators connected  with  pumps  in  the  same  phase  have  hemispheres 
of  the  same  color  advancing  simultaneously. 

14.  Instrument  for  the  Registering  of  ]rater  Oseillations. — 
When  a  pulsating  or  an  oscillating  body,  like  one  of  those  just  de- 
scribed, is  placed  in  the  water,  the  motion  produced  by  it  cannot 
be  seen,  as  an  obvious  consequence  of  the  transparency  of  the 
water.  This  motion  can,  however,  be  observed  indirectly  in 
several  ways.  For  example,  we  can  suspend  small  particles  in 
the  water  and  observe  their  motions,  and  we  might  even  succeed 
in  getting  photographs  of  the  paths  of  oscillation  of  the  suspended 

3 


18  FIELDS    OF    FORCE. 

particles.  This  method  has,  however,  never  been  nsed,  and  may 
involve  difficulties  because  of  the  small  amplitudes  of  the  oscilla- 
tions. 

A  more  mechanical  method,  depending  upon  the  principle  of 
kinetic  buoyancy,  is  preferable.  A  body  which  is  situated  in  the 
oscillating  masses  of  fluid  will  be  subject  to  a  periodic  kinetic 
buoyancy  which  tries  to  set  up  in  it  oscillations  of  the  same  direc- 
tion as  those  of  the  water.  The  amplitudes  of  the  oscillations 
produced  will,  however,  generally  be  minute,  but  they  may  be  in- 
creased by  resonance.  The  body  is  fixed  upon  an  elastic  wire,  and 
the  period  of  the  generator  varied  until  it  accords  with  the  period 
of  the  free  vibrations  of  the  body.  The  amplitude  of  the  oscilla- 
tions of  the  body  is  then  greatly  increased. 

The  body  is  made  to  carry  a  hair  pencil,  which  reaches  above  the 
surface  of  the  water.  One  or  two  millimeters  above  the  point  of 
the  brush  is  placed  a  horizontal  glass  plate,  resting  upon  springs. 
When  the  body  has  acquired  large  oscillations,  the  glass  plate  may 
be  pressed  down  and  the  brush  marks  an  ink  line  upon  it.  The 
registering  device  is  then,  moved  to  another  place  in  the  fluid,  and 
the  direction  of  the  water  oscillations  at  this  place  recorded  on  the 
glass  plate,  and  so  on.  In  this  way  comj)lete  diagrams  of  the  lines 
of  oscillation  in  the  fluid  are  obtained. 

15.  T)i<((/rani.s  of  lli/drodynamic  and  Corresijondmg  Magnetic 
Fields. — Figs.  4-8,  «,  give  diagrams  of  hydrodynamic  fields  ob- 
tiiined  in  this  way,  while  Figs.  4-8,  b,  give  the  diagrams  of  the 
corresponding  magnetic  fields,  obtained  in  the  well  known  way 
with  iron  filings. 

Fig.  4,  a,  gives  the  radial  lines  of  oscillation  obtained  in  the 
space  around  a  pulsating  body,  while  Fig.  4,  b,  gives  the  corre- 
sponding magnetic  lines  of  force  issuing  from  one  pole  of  a  long 
bar  magnet. 

Fig.  0,  a,  gives  the  lines  of  oscillation  produced  in  the  fluid  by 
two  bodies  pulsating  in  the  same  phase.  They  represent  the  meet- 
ing of  two  radial  currents  issuing  from  two  centers.  Fig.  5, 
b,  gives  the  perfectly  analogous  representation  of  the  magnetic  lines 
of  force  issuing  from  two  magnetic  poles  of  the  same  sign. 


INV 


.^ESTIGATION    OF    GEOMETRIC    PROPERTIES. 


19 


-      ^     I     ;       /      / 


\ 


O 


/ 


r-/  / 


/   /  :  \  \  \ 

/      /       !     '     ^ 


b 

Fig.  4. 


Fi.  6  «,  gives  the  lines  of  oscillation  produced  iu  the  fluid  by 
two  bodi;s  iatiug  iu  opposite  phase.  The  <^'^^2^^ 
representation  of  a  current  which  diverges  from  one  pulsatn,g  bodj 


20 


FIELDS   OF    FORCE. 


and  converges  toward  the  other.  Fig.  6,  6,  gives  the  perfectly 
analogous  representation  of  the  magnetic  lines  of  force  produced 
bv  two  magnetic  poles  of  opposite  sign. 


i     J      '     I    f         M   \      ( 


I      / 


\  \  \  ;  ;  /  /  i-'ii  \^  \  ;  '  '  '  /  /  / 


/  /,/ 


o 


//;//  \  \  \V:','//// 


^^:^^ 


1  \  \  ^ 


b 

Fio.  5. 


^ 'g-  ">  ''>  gives  the  more  complicated  representation  of  the  line 
of  oscillation  produced  in  tiio  water  l)ya  combination  of  three  pul- 


INVESTIGATION    OF    GEOMETRIC    PROPERTIES. 


21 


sating  bodies,  two  ])ul.sating  in  the  same  phase,  and  one  in  the 
opposite,  and  Fig.  7,  h,  gives  the  perfectly  analogous  representa- 
tion of  the  magnetic  lines  of  force  produced  by  three  magnetic 
poles,  of  which  two  have  the  same  sign,  and  one  the  opposite. 


I     /      /     . 

/    / 


/    / 


o 


.^^. 


V'-:///,i,\A>:r-'f^!,\\ 


t  \ 


Finally,  Fig.  8,  «,  giv^es  the  lines  of  oscillation  produced  in  the 
fluid  by  an  oscillating  body,  and  Fig.  8,  b,  the  corresponding  lines 
of  magnetic  force  produced  by  a  short  magnet. 


22 


X 


\\M 


FIELDS    OF    FORCE. 


/ 
X 


T--V/'" 


/       \    \  \\  \\\\V\     /  /  /  /   /        »     ^ 


■^ 


/  iW 


INVESTIGATION    OF    GEOMETRIC    PROPERTIES. 


23 


These  figures  show  very  fully  the  analogy  in  the  geometry  of 
the  fields  produced,  on  the  one  hand,  by  magnetic  poles  or  magnets 
in  a  surrounding  homogeneous  medium,  and,  on  the  other  hand. 


by  pulsating  or  oscillating  bodies  in  a  surrounding  homogeneous 
fluid.     The  experimental  demonstration  of  the  analogy  for  the  case 


24  FrKI.DS    OK    FOUCK. 

wlieii  tlic  iiu'ditini  siirrouiuling  the  magnets  and  the  fluid  siirround- 
iiifi;  thepidsatiiigor  oscilhitinsi^  bodies  con  tain  heterogeiu'itie.sis  more 
delicate.  In  the  hydrodyntunic  ease  the  heterogeneities  should  be 
Ihiid,  and  it  is  |)ra(!tieally  iinpossil)le,  on  account  of  the  action  of 
t;ravity,  to  have  a  fluid  mass  of  given  shape  flowing  freely  in  a 
fluid  of  other  density.  If  for  the  fluid  bodies  we  substitute  rigid 
bodies,  suspended  from  above  or  anchored  from  below,  according  to 
(heir  density,  it  is  easily  seen,  by  means  of  our  registering  device, 
tliat  the  lines  of  oscillation  have  a  tendency  to  converge  toward  the 
light,  and  to  diverge  from  the  heavy  bodies.  But  this  registering 
device  cannot  be  brought  sufficiently  near  these  bodies,  to  show 
the  curves  in  their  immediate  neighborhood.  Here  the  observa- 
tion of  the  oscillations  of  small  suspended  ])artieles  would  probably 
be  the  best  method  to  employ.  Experiments  which  we  shall  per- 
form later  will  give,  however,  indire(!t  proofs  that  the  fields  have 
exactly  the  expected  character. 

16.  On  PossiUe  Kxfcnsioits  of  tlic  Aualogy.  —  We  have  thus 
found,  by  elementary  reasoning,  a  very  complete  analogy  between 
the  geometric  propertiqs  of  hydrodynamic  fields  and  electric  or 
magnetic  flelds  for  the  case  of  statical  })henomena.  And,  to  s'ome 
extent,  we  have  verified  these  results  by  experiments. 

It  is  a  natural  (|uestion  then,  does  the  analogy  extend  to  fields 
of  greater  generality,  or  to  flelds  of  electromagnetism  of  the  most 
general  nature?  In  discussing  this  question  further  an  introduc- 
tory remark  is  important.  The  formal  analogy  which  exists  be- 
tween electrostatic  and  magnetic  fields  has  made  it  possible  for 
us  to  compare  the  hydrodynamic  flelds  considered  with  both  elec- 
trostatic and  magnetic  fields.  If  there  exists  a  perfect  hydro- 
dynamic  analogy  to  electromagnetic  phenomena,  the  liydrodynamic 
fields  considered  will,  presiimal)ly,  turn  out  to  be  analogous  either 
to  electrostatic  fields  only,  or  to  magnetic  fields  only,  but  not  to 
both  at  the  same  time.  The  (piestion  therefore  can  now  be  raised, 
would  our  hydrodynamic  fields  in  an  eventually  extended  analogy 
correspond  to  the  electrostatic  or  the  magnetic  flelds?  To  this  it 
must  be  answered,  it  is  very  probable  that  only  the  analogy  to  the 


IXVKSTKiATIOX    OF    C  KOM  irriMC    I'UOl'Kiri'I  KS.  25 

electrostatic  fields  will  hoUl.  As  an  obvious  argument,  it  may  be 
eiui)liasize(l  that  the  hydrodynamic  fields  have  exactly  the  gener- 
ality of  electrostatic  fields,  but  greater  generality  than  magnetic 
fields.  The  analogy  to  magnetism  will  take  the  right  form  only 
when  the  restriction  is  introduced,  that  changes  oi'  volume  are  to  be 
excluded.  Otherwise,  we  should  arrive  at  a  theory  of  magnetism 
where  isolated  magnetic  poles  could  exist.  To  this  argument 
others  may  be  added  later. 

But  in  spite  of  this,  the  formal  analogy  of  the  electric  and  mag- 
netic fields  makes  it  possible  to  formally  compare  hydrodynamic 
fields  with  magnetic  fields.  And  this  will  often  be  preferable,  fi)r 
practical  reasons.  This  will  be  the  case  in  the  following  discus- 
sion, because  the  idea  of  the  electric  current  is  much  more  familiar 
to  us  than  the  idea  of  the  magnetic  current,  in  spite  of  the  formal 
analogy  of  these  two  currents. 

Let  us  compare,  then,  the  hydrodynamic  fields  hitherto  consid- 
ered with  magnetic  fields  produced  by  steel  magnets.  The  lines  of 
force  of  these  fields  always  pass  through  the  magnets  which  produce 
them,  just  as  the  corresponding  hydrodynamic  curves  pass  through 
the  moving  bodies  which  produce  the  motion.  The  magnetic  lines 
of  force  produced  by  electric  currents,  on  the  other  hand,  are  gener- 
ally closed  in  the  exterior  space,  and  need  not  pass  at  all  through 
the  conductors  carrying  the  currents.  To  take  a  simple  case,  the 
lines  of  force  produced  by  au  infinite  rectilinear  current  are  circles 
around  the  current  as  an  axis. 

If  it  should  be  possible  to  extend  the  analogy  so  as  to  include 
also  the  simplest  electromagnetic  fields,  we  would  have  to  look  for 
hydrodynamic  fields  with  closed  lines  of  fiow  which  do  not  ])ass 
through  the  bodies  ]u-oducing  the  motion.  It  is  easily  precon- 
ceived, that  if  the  condition  of  the  oscillatory  nature  of  the  fiuid 
motion  be  insisted  upon,  the  recpiircd  motion  cannot  be  pro- 
duced by  fluid  pressure  in  a  perfect  fiuid.  A  cylinder,  for 
instance,  making  rotary  oscillations  around  its  axis  will  produce 
no  motion  at  all  in  a  perfect  fluid,  (^uite  the  contrary  is  true, 
if  the  fluid  be  viscous,  or  if  it  have  a  suitable  transverse  elasticity, 

4 


26 


FIELDS    OF    FORCE. 


as  does  an  aqueous  solution  of  gelatine.  But,  as  we  shall  limit 
ourselves  to  the  consideration  of  perfect  fluids,  we  shall  not  con- 
sider the  phenomena  in  such  media. 

17.  Detached  Ui/drodynamic  Ancdogy  to  the  Fields  of  Stationary 
Electromagnetism. — A  direct  continuation  of  our  analogy  is  thus 
made  impossible.  It  is  a  very  remarkable  fact,  however,  that 
there  exist  hydrodynamic  fields  which  are  geometrically  analogous  to 
the  fields  of  stationary  electric  currents.  But  to  get  these  fields 
we  must  give  up  the  condition,  usually  insisted  upon,  that  the 
motion  be  of  oscillatory  nature.  We  thus  arrive  at  an  inde- 
pendent analogy,  which  has  a  considerable  interest  in  itself,  but 
which  is  no  immediate  continuation  of  that  considered  above. 


Fi(i.  9. 

Tliis  analogy  is  that  discovered  by  v.  Helmholtz  in  his 
research  (»n  the  vortex  motion  of  perfect  fluids.  According  to  his 
celebrated  results,  a  vortex  can  be  compared  with  an  electric  cur- 
rent, and  the  fluid  field  surrounding  the  vortex  will  then  be  in 
exactly  the  same  relation  to  the  vortex  as  the  magnetic  field  is 
to  the  electric  current  which  produces  it. 

To  consider  only  the  case  of  rectilinear  vortices,  the  field  of  one 
rectilinear  vortex  is  represented  by  concentric  circles.  And  this 
field  corresponds  to  the  magnetic  field  of  a  rectilinear  current. 
The  hydrodynamic  field  of  two  rectilinear  parallel  vortices  which 


INVESTIGATION    OF    GEO]\[ETRIC    PROPERTIES. 


27 


have  the  same  direction  of  rotation  is  shown  in  Fig.  9,  and  this 
field  is  strictly  analogous  to  the  magnetic  field  of  two  rectilinear 
parallel  currents  in  the  same  direction.  Fig.  10  gives  the  hydro- 
dynamic  field  of  two  rectilinear  parallel  vortices  which  have 
opposite  directions  of  rotation,  and  it  is  strictly  analogous  to  the 
magnetic  field  of  two  electric  currents  of  opposite  direction. 

Fields  of  this  nature  can  be  easily  produced  in  water  by  rotat- 
ing rigid  cylinders,  and  observed  by  the  motion  of  suspended  par- 
ticles. At  the  same  time,  each  cylinder  forms  an  obstruction  in 
the  field  produced  by  the  other.     If  only  one  cylinder  be  rotating, 


Fig.  10. 

the  lines  of  flow  produced  by  it  will  be  deflected  so  that  they  run 
tangentially  to  the  surface  of  the  other.  The  cylinder  at  rest  thus 
influences  the  field  just  as  a  cylinder  of  infinite  diamagnetivity 
would  influence  the  magnetic  field.  The  rotating  cylinders  there- 
fore correspond  to  conductors  for  electric  currents,  which  are  con- 
structed in  a  material  of  infinite  diamagnetivity. 

This  analogy  to  electromagnetism  is  limited  in  itself,  apart 
from  its  divergence  from  the  analogy  considered  previously. 
The  extreme  diamagnetivity  of  the  bodies  is  one  limitation.     An- 


"2S  FIELDS    OF    FORCE. 

other  limitation  follows  from  Helmholtz's  celebrated  theorem, 
that  vortices  do  not  vary  in  intensity.  Therefore  phenomena 
corresponding  to  those  of  electromagnetic  induction  are  excluded. 
Whichever  view  we  take  of  the  subject,  the  hydrodynamic 
analogies  to  electric  and  magnetic  phenomena  are  thus  limited  in 
extent.  To  get  analogies  of  greater  extent  it  seems  necessary  to 
pass  to  media  witii  other  properties  than  those  of  perfect  fluids. 
But  we  will  not  try  on  this  occasion  to  look  for  further  exten- 
sions of  the  geometric  analogies.  We  prefer  to  pass  to  an  exami- 
nation of  the  dynamic  properties  of  tiie  fields  whose  geometric 
properties  we  have  investigated. 


II. 

ELEMENTARY  INVESTIGATION  OF  THE  DYNAMI- 
CAL PROPERTIES  OF  HYDRODYNAMIC  FIELDS. 

1.  The  Dynamics  of  the  Electric  or  the  Magnetic  Field  — Our 
knowledge  of  the  dynamics  of  the  electric  or  magnetic  field  is 
very  incomplete,  and  will  presumably  remain  so  as  long  as  the 
true  nature  of  the  fields  is  unknown  to  us. 

What  we  know  empirically  of  the  dynamics  of  the  electric  or 
magnetic  field  is  this  — bodies  in  the  fields  are  acted  upon  by 
forces  which  may  be  calculated  when  we  know  the  geometry  of 
the  field.  Under  the  influence  of  these  forces  the  bodies  may 
take  visible  motions.  But  we  have  not  the  slightest  idea  of  the 
hidden  dynamics  upon  which  these  visible  dynamic    phenomena 

depend. 

Faraday's  idea,  for  instance,  of  a  tension  parallel  to,  and  a 
pressure  perpendicular  to  the  lines  of  force,  as  well  as  Maxwell's 
mathematical  translation  of  this  idea,  is  merely  hypothetical. 
And  even  though  this  idea  may  contain  more  or  less  of  the 
truth,  investigators  have  at  all  events  not  yet  succeeded  in  mak- 
ing this  dynamical  theory  a  central  one,  from  which  all  the 
properties  of  the  fields,  the  geometric,  as  well  as  the  dynamic, 
naturally  develop,  just  as,  for  example,  all  properties  of  hydro- 
dynamic  fields,  the  geometric,  as  well  as' the  dynamic,  develop 
from  the  hydrodynamic  equations.  Maxwell  himself  was  very 
well  aware  of  this  incompleteness  of  his  theory,  and  he  stated  it 
in  the  following  words  : 

"  It  must  be  carefully  born  in  mind  that  we  liave  only  made 
one  step  in  the  theory  of  the  action  of  the  medium.     We  have 
supposed  it  to  be  in  a  state  of  stress  but  have  not  in  any  way  ac- 
counted for  this  stress,  or  explained  how  it  is  maintained.   .   .   . 
"  I  have  not  been  able  to  make  the  next  step,  namely,  to  ac- 

29 


30  FIELDS   OF   FORCE. 

count  by  mechanical  considerations  for  these  stresses  in  the  di- 
electric." 

In  spite  of  all  formal  progress  in  the  domain  of  Maxwell's 
theory,  these  words  are  as  true  to-day  as  they  were  when  Max- 
AVELL  wrote  them.  This  circumstance  makes  it  so  much  the  more 
interesting  to  enter  into  the  dynamic  properties  of  the  hydrody- 
namic  fields,  which  have  shown  such  remarkable  analogy  in  their 
geometric  properties  to  the  electric  or  magnetic  fields,  in  order  to 
see  if  with  the  analogy  in  the  geometric  properties  there  will  be 
associated  analogies  in  their  dynamical  properties.  The  question 
is  simply  this  : 

Consider  an  electric,  or  magnetic  field  and  the  geometrically 
corresponding  hydrodynamic  field.  Will  the  bodies  which  pro- 
duce the  hydrodynamic  field,  namely,  the  pulsating  or  the  oscillat- 
ing bodies  or  the  bodies  which  modify  it,  such  as  bodies  of  other 
density  than  the  surrounding  fluid,  be  subject  to  forces  similar 
to  those  acting  on  the  corresponding  bodies  in  the  electric  or 
magnetic  fields  ? 

This  question  can  be  answered  by  a  simple  application  of  the 
principle  of  kinetic  buoyancy. 

2.  Jiesu/faiit  Force  against  a  Pidmting  Body  in  a  Si/nchronoudi/ 
Oscillatiiir/  Cun'ent.  —  Let  us  consider  a  body  in  the  current  pro- 
duced by  any  system  of  synchronously  pulsating  and  oscillating 
bodies.  It  will  be  continually  subject  to  a  kinetic  buoyancy  pro- 
portional to  the  product  of  the  acceleration  of  tlie  fluid  masses  into 
the  mass  of  water  displaced  by  it.  If  its  volume  be  constant,  so 
that  the  displaced  mass  of  water  is  constant,  the  force  will  be 
strictly  i)eriodic,  with  a  mean  value  zero  in  the  period.  It  will 
then  be  brought  only  into  oscillation,  and  no  progressive  motion 
will  result. 

Jiut  if  the  body  has  a  variable  volume,  the  mass  of  water  dis- 
placed by  it  will  not  be  constant.  If  the  changes  of  volume  con- 
sist in  |)ulsation.s,  synchronous  with  the  pulsations,  or  oscillations, 
of  the  distant  bodies  which  produce  the  current,  the  displaced 
mass  of  water  will  have  a  maximum  when  the  acceleration  has  its 


INVESTIGATION    OF    DYNAMICAL    PROPERTIES.  31 

nuixiinum  in  one  direction,  and  a  minimum  when  the  acceleration 
lias  its  maximum  in  the  opposite  direction.  As  is  seen  at  once, 
the  force  can  then  no  longer  have  the  mean  value  zero  in  the  period. 
It  will  have  a  mean  value  in  the  direction  of  the  acceleration  at  the 
time  when  the  pulsating  body  has  its  maximum  volume.  We  thus 
find  the  result : 

A  pulsating  body  in  a  si/nchronousli/  osciUating  current  is  subject 
to  the  action  of  a  resultant  force,  the  direction  of  which  is  that  of  the 
acceleration  in  the  current  at  the  time  ichen  the  pulsating  body  has  its 
maximum  volume. 

3.    ITutual  Attraction  and  Bepidsion    between    Two    Pulsating 

Bodies. As  a  first  application  of  this  result,  we  may  consider  the 

case  of  two  synchronously  pulsating  bodies.  Each  of  them  is  in 
the  radial  current  produced  by  the  other,  and  we  have  only  to 
examine  the  direction  of  the  acceleration  in  this  current.  Evi- 
dently, this  acceleration  is  directed  outwards  when  the  body  pro- 
ducing it  has  its  minimum  volume,  and  is  therefore  about  to  expand, 
and  is  directed  inwards  when  the  body  producing  it  has  its  maxi- 
mum volume,  and  is  therefore  about  to  contract. 

Let  us  consider  first  the  case  of  two  bodies  pulsating  in  the 
same  phase.  They  have  then  simultaneously  their  maximum  vol- 
umes, and  the  acceleration  in  the  radial  current  produced  by  the 
one  body  will  thus  be  directed  inwards,  as  regards  itself,  when  the 
other  body  has  its  maximum  volume.  The  bodies  will  therefore 
be  driven  towards  each  other ;  there  will  be  an  apparent  mutual 
attraction.  If,  on  the  other  hand,  the  bodies  pulsate  in  opposite 
phase,  one  will  have  its  maximum  volume  when  the  other  has  its 
minimum  volume.  And  therefore  one  will  have  its  maximum  vol- 
ume when  the  radial  acceleration  is  directed  outward  from  the 
other.  The  result,  therefore,  will  be  an  apparent  mutual  repulsion. 
As  the  force  is  proportional  to  the  acceleration  in  the  radial  cur- 
rent,and  as  the  acceleration  will  decrease  exactly  as  the  velocity,  pro- 
portionally to  the  inverse  square  of  the  distance,  the  force  itself 
will  also  vary  according  to  this  law.  On  the  other  hand,  it  is 
easily  seen  that  the  force  must  also  be  proportional  to  two  param- 


32  FIEI.DS    OF    FORCE. 

eters,  which  measure  in  a  proper  way  the  intensities  of  the  pulsa- 
tions of  each  body.  Calling  these  parameters  the  "intensities  of 
pulsation,"  we  find  the  following  law  : 

Between  bodies  puhating  in  the  same  phase  there  is  an  apparent 
attraction;  between  bodies  pulsating  in  the  opposite  phase  there  is  an 
apparent  repulsion,  the  force  being  proportional  to  the  product  of  the 
two  intensities  of  pulsation,  and  proportional  to  the  inverse  square  of 
the  distance. 

4.  Discussion. — "We  have  thus  deduced  from  the  principle  of 
dynamic  buoyancy,  that  is  from  our  knowledge  of  the  dynamics 
of  the  hydrodynamic  field,  that  there  will  be  a  force  which  moves 
the  pulsating  bodies  through  the  field,  just  as  there  exists,  for 
reasons  unknown  to  us,  a  force  which  moves  a  charged  body 
through  the  electric  field.  And  the  analogy  is  not  limited  to  the 
mere  existence  of  the  force.  For  the  law  enunciated  above  has 
exactly  the  form  of  Coulojik's  law  for  the  action  between  two 
electrically  charged  particles,  with  one  striking  difference ;  the 
direction  of  the  force  in  the  hydrodynamic  field  is  opposite  to  that 
of  the  corresponding  force  in  the  electric  or  magnetic  field.  For 
bodies  pulsating  in  the  same  phase  must  be  compared  wdth  bodies 
charged  with  electricity  of  the  same  sign  ;  and  bodies  pulsating 
in  the  opposite  phase  must  be  compared  with  bodies  charged  with 
opposite  electricities.  This  follows  inevitably  from  the  geometrical 
analogy.  For  bodies  pulsating  in  the  same  phase  produce  a  field 
of  the  same  geometrical  configuration  as  bodies  charged  with 
the  same  electricity  (Fig.  5,  a  and  b) ;  and  bodies  pulsating  in 
opjwsite  i)hase  produce  the  same  field  as  bodies  charged  with 
opposite  electricities  (Fig.  6,  a  and  6). 

This  exception  in  the  otherwise  complete  analogy  is  most  aston- 
ishing, liut  we  cannot  discover  the  reason  for  it  in  the  present 
limited  state  (»f  our  knowledge.  We  know  very  well  why  the 
force  in  the  hydrodynamic  field  must  have  the  direction  indicated 
—  this  is  a  simple  consequence  of  the  dynamics  of  the  fluid.  But 
in  our  total  ignorance  of  the  internal  dynamics  of  the  electric  or 
magnetic  field  we  cannot  tell  at  all  why  the  force  in  the  electric 
field  has  the  direction  which  it  has,  and  not  the  reverse. 


INVESTIGATION    OF    DYNAMICAL    PROPERTIES.  33 

Thus,  taking  the  focts  as  we  find  them,  we  arrive  at  the  result 
that  with  the  geometrical  analogy  developed  in  the  preceding  lec- 
ture there  is  associated  an  inverse  dynamical  analogy  : 

Falsating  bodies  act  upon  each  other  as  if  they  were  electrically 
charged  partides  or  magnetic  jjoles,  but  with  the  difference  that 
charges  or  poles  of  the  same  sign  attract,  and  charges  or  i^oles  of 
opposite  sign  repel  each  other. 

5.  Pulsation  Balance. — In  order  to  verify  this  result  by  experi- 
ment an  arrangement  must  be  found  by  which  a  pulsating  body 
lias  a  certain  freedom  to  move.  This  may  be  obtained  in  different 
ways.  Thus  a  pulsator  may  be  suspended  as  a  pendulum  by  a 
long  india-rubber  tube  through  which  the  air  from  the  generator 
is  brought.  Or  it  may  be  inserted  in  a  torsion  balance,  made  of 
glass  or  metal  tubing,  and  suspended  by  an  india-rubber  tube  which 
brings  the  air  from  the  generator  and  at  the  same  time  serves  as  a 
torsion  wire.  These  simple  arrangements  have  at  the  same  time 
the  advantage  that  they  allow  rough  quantitative  measurements  of 
the  force  to  be  made.  For  good  qualitative  demonstrations  the 
following  arrangement  will  generally  be  found  preferable. 

The  air  from  the  generator  comes  through  the  horizontal  metal 
tube,  a,  (Fig.  11),  which  is  fixed  in  a  support.  The  air  channel 
continues  vertically  through  the  metal  piece  b,  which  has  the  form 
of  a  cylinder  with  vertical  axis.  At  the  top  of  this  metal  piece 
and  in  the  axis  there  is  a  conical  hole,  and  the  lower  surface 
is  spherical  with  this  hole  as  center.  The  movable  part  of  the 
instrument  rests  on  an  adjustable  screw,  pivoted  in  this  hole.  This 
screw  carries,  by  means  of  the  arm  d,  the  little  cylinder  c,  through 
which  the  vertical  air  channel  continues.  The  upper  surface  of  this 
cylinder  is  spherical,  with  the  point  of  the  screw  as  center.  The 
two  spherical  surfaces  never  touch  each  other,  but  by  adjustment 
of  the  screw  they  may  be  brought  so  near  each  other  that  no  sensi- 
ble loss  of  air  takes  place.  To  the  part  of  the  instrument  c-d, 
which  gives  freedom  of  motion,  the  pulsator  may  be  connected  by 
the  tube  ef,  the  counter-weight  maintaining  the  equilibrium.  By 
this  arrangement,  the  pulsating  body  is  free  to  move  on  a  spherical 
5 


34 


FIEI.DS    OF    FORCE. 


surface  with  the  pivot  as  center,  and  the  equilibrium  will  be  neutral 
for  a  horizontal  motion,  and*  stable  for  a  vertical  motion. 

G.   ExperimenU   with  Pulsating  Bodies. — Having   one  pulsator 
in  the  pulsation  balance,  take  another  in  the  hand,  and  arrange  the 


Fig.  11. 


generator  for  ])alsations  of  the  same  phase,  and  we  see  at  once  that 
the  two  pulsating  bodies  attract  each  other  (Fig.  12,  a).  This 
attraction  is  easily  seen  with  distances  up  to  10^15  cm.,  or  more, 
and  it  is  observed  that  the  intensity  of  the  force  increases  rapidly 


Fig.  12. 


as  the  distance  diminishes.  The  moment  the  relative  phase  of  the 
pulsations  is  changed,  the  attraction  ceases,  and  an  equally  intense 
rc|)ii]sion  appears  (Fig.  12,  I>).     With  the  torsion  balance  it  may 


INVESTIGATION    OF    DYNAMICAL    PROPERTIES.  35 

be  shown  with  tolerable  accuracy,  that  the  force  varies  as  the  in- 
verse square  of  the  distance,  and  is  proportional  to  two  parameters, 
the  intensities  of  pulsation. 

In  this  experiment  the  mean  value  only  of  the  force  and  the 
progressive  motion  produced  by  it  are  observed.  By  using  very 
slow  pulsations  with  great  amplitudes,  a  closer  analysis  of  the  phe- 
nomenon is  possible.  It  is  then  seen  that  the  motion  is  not  a 
simple  progressive  one,  but  a  dissymmetric  vibratory  motion,  in 
which  the  oscillations  in  the  one  direction  always  exceed  a  little 
the  oscillations  in  the  other,  so  that  the  result  is  the  observed 
progressive  motion. 

7.  Actio)i  of  an  Oscillating  Body  upon  a  Pulsating  Body.  — 
Two  oppositely  pulsating  bodies  produce  geometrically  the  same 
field  as  two  opposite  magnetic  poles.  Geometrically,  the  field  is 
that  of  an  elementary  magnet.  Into  the  field  of  these  two  oppo- 
sitely pulsating  bodies  we  can  bring  a  third  pulsating  body. 
Then,  if  we  bring  into  application  the  law  just  found  for  the 
action  between  two  pulsating  bodies,  we  see  at  once  that  the  third 
pulsating  body  will  be  acted  upon  by  a  force,  opposite  in  direc- 
tion to  the  corresponding  force  acting  on  a  magnetic  pole  in 
the  field  of  an  elementary  magnet.  In  this  result  nothing  will 
be  changed,  if,  for  the  two  oppositely  pulsating  bodies,  we  substitute 
an  oscillating  body.  For  both  produce  the  same  field,  and  the 
action  on  the  pulsating  body  will  evidently  depend  only  upon  the 
field  produced,  and  not  upon  the  manner  in  which  it  is  produced. 
We  thus  find  : 

An  oscillating  body  will  act  upon  a  pulsating  body  as  an  ele- 
mentary magnet  upon  a  magnetic  pole,  but  tcith  the  kmo  of  poles 
7-eversed. 

This  result  may  be  verified  at  once  by  experiment.  If  we  take 
an  oscillator  in  the  hand,  and  bring  it  near  the  pulsator  which  is 
inserted  in  the  pulsation-balance,  we  find  attraction  in  the  case 
(Fig.  13,  rt)  when  the  oscillating  body  approaches  the  pulsating 
body  as  it  expands  and  recedes  from  it  as  it  contracts.  But  as 
soon  as  the  oscillating  body  is  turned  around,  so  that  it  approaches 


36 


FIELDS    OF    FORCE. 


while  the  pulsating  body  is  contracting  and  recedes  while  it  is 
expanding  (Fig.  13,6),  the  attraction  changes  to  repulsion. 

To  show  how  the  analogy  to  magnetism  goes  even  into  the 
smallest  details  the  oscillating  body  may  be  placed  in  the  })ro- 
longation  of  the  arm  of  the  pulsation-balance,  so  that  its  axis 
of  oscillation  is  perpendicular  to  this  arm.  The  pulsating  body 
will  then  move  a  little  to  one  side  and  come  into  equilibrium  in  a 
dissymmetric  position  on  one  side  of  the  attracting  pole  (Fig. 
13,  f).  If  the  oscillating  body  be  turned  around,  the  position  of 
equilibriiuii   will   be  on  the  other  side.      Exactly  the  same  small 


Fig.  13. 

lateral  displacement  is  observed  when  a  short  magnet  is  brought 
into  the  transverse  position  in  the  neighborhood  of  the  pole  of 
a  l(»ng  bar  magnet  which  has  the  same  freedom  to  move  as  the 
l)ulsating  body. 

8.  Force  mjaind  an  Omillatiinf/  Body.  —  If,  in  the  preceding 
experiment,  we  take  the  pulsating  body  in  the  hand  and  insert  the 
oscillating  body  in  the  balance,  we  cannot  conclude  a  ■priori  that 
the  motions  of  the  oscillating  body  will  prove  the  existence  of 
a  force  ccpial  and  opposite  to  that  exerted  by  the  oscillating  body 
upon  the  i)ulsating  body.  The  principle  of  equal  action  and  re- 
action is  empirically  valid  for  the  common  actions  at  a  distance 
between  two  bodies.  But  for  these  apparcnf  actions  at  a  distance, 
where  not  only  the  two  bodies  but  also  a  third  one,  the  fluid,  are 
engaged,  no  general  coriclusion  can  l)e  drawn. 


INVESTIGATION    OF    DYNAMICAL    PROPERTIES. 


37 


To  examine  the  action  to  which  the  oscillating  body  is  subject 
we  must  therefore  go  back  to  the  principle  of  kinetic  buoyancy. 
The  kinetic  buoyancy  will  give  no  resultant  force  against  a  body 
of  invariable  volume,  which  oscillates  between  two  places  in 
the  fluid  where  the  motion  is  the  same.  For  at  both  ends  of 
the  path  the  body  will  be  subject  to  the  action  of  equal  and  oppo- 
site forces.  But  if  it  oscillates  between  places  where  the  motion 
is  somewhat  different  in  direction  and  intensity,  these  two  forces 
will  not  be  exactly  equal  and  opposite.  The  direction  of  the 
accelerations  in  the  oscillating  fluid  masses  is  always  tangential 
to  the  lines  of  oscillation.  If  the  field  be  represented  by  these 
lines,  and  if  the  absolute  value  of  the  acceleration  be  known  at 
every  point  of  the  fluid  at  any 
time,  the  force  exerted  on  the 
oscillating  body  at  every  point 
of  its  path  may  be  plotted,  and 
the  average  value  found.  As 
we  desire  only  qualitative  re- 
sults, it  will  be  sufficient  to 
consider  the  body  in  the  two 
extreme  positions  only,  where 
we  have  to  do  with  the  ex- 
treme   values    of     the    force. 

Let,  then,  the  continuous  circle  (Fig.  14)  represent  the  oscillat- 
ing body  in  one  extreme  position,  and  the  dotted  circle  the  same 
body  in  the  other  extreme  position,  and  let  the  two  arrows  be  pro- 
portional to  the  accelerations  which  the  fluid  has  at  these  two  places 
at  the  corresponding  times.  The  composition  of  these  two  alter- 
nately acting  forces  gives  the  average  resultant  force.  Let  us  now 
substitute  for  the  oscillating  body  a  couple  of  oppositely  pulsating 
bodies,  one  in  each  extreme  position  of  the  oscillating  body,  and 
let  us  draw  arrows  representing  the  average  forces  to  which  these 
two  pulsating  bodies  are  subject.  We  then  get  arrows  located 
exactly  as  in  the  preceding  case.  And  we  conclude,  therefore, 
that  if  we  only  adjust  the  intensities  of  pulsation  properly,  the 


38  FIELDS    OF    FORCE. 

two  oppositely  pulsating  bodies  will  be  acted  upon  by  exactly  the 
same  average  resultant  force  as  the  oscillating  body.  From  the 
results  found  above  for  the  action  against  pulsating  bodies  we  can 
then  conclude  at  once  : 

An  oscUlatiiif/  body  in  the  hijdi-odynamic  field  vnll  be  subject  to  the 
action  of  d  force  similar  to  that  acting  upon  an  elementary  magnet 
in  the  magnetic  field,  the  onhj  difference  being  the  difference  in  the 
signs  of  the  forces  which  follows  from  the  opposite  joole-law. 

9.  Experimental  Investigation  of  the  Force  exerted  by  a  Pulsat- 
ing Body  upon  an  Oscillatiny  Body.  —  Let  us  now  insert  the 
oscillator  in  the  balance,  and  turn  it  so  that  the  axis  of  oscillation 
is  in  the  direction  of  its  free  movement.  If  a  pulsator  be  taken 
in  the  hand,  it  will  be  seen  that  attraction  takes  place  when  the 
pulsiiting  body  is  made  to  approach  one  pole  of  the  oscillating 
body  (Fig.  13,  a),  and  repulsion  if  it  is  made  to  approach  the 
other  pole  (Fig,  13,  b).  And,  as  is  evident  from  comparison  with 
the  preceding  case,  the  force  acting  on  the  oscillating  body  is  al- 
ways opposite  to  that  acting  on  the  pulsating  body.  We  have 
equality  of  action  and  reaction,  just  as  in  the  case  of  magnetism. 

The  analogy  with  magnetism  can  be  followed  further  if  the 
pulsating  i)ody  be  brought  into  the  prolonged  arm  of  the  oscilla- 
tion balance.  The  oscillating  body  will  then  take  a  short  lateral 
displacement,  so  that  its  attracting  pole  comes  nearer  to  the  pul- 
sating body  (Fig.  13,  c).  It  is  a  lateral  displacement  correspond- 
ing exactly  to  that  take  by  an  elementary  magnet  under  the  influ- 
ence of  a  magnetic  pole. 

10.  Experimental  Investigation  of  the  Mutual  Actions  betiveen 
Two  Oscillating  llodicx.  —  The  pulsator  held  in  the  hand  may  now 
be  replaced  by  an  oscillator,  while  the  oscillator  inserted  in  the 
balance  is  left  unchanged,  so  that  it  is  still  free  to  move  along  its 
axis  of  oscillation.  We  may  first  bring  the  oscillator  held  in  the 
hand  into  the  position  indicated  by  the  figures  15,  a  and  6,  so  that 
tlie  axes  of  osciUation  lie  in  the  same  line.  The  experiment  will 
then  correspond  to  that  with  magnets  in  longitudinal  position.  We 
get  attraction  in  the  case,  (Fig.  15,  a),  when  the  oscillating  bodies 


INVESTIGATION    OF    DYNAMICAL    PROPERTIES. 


39 


are  in  opposite  phase.  This  corresponds  to  the  case  in  which  the 
magnets  have  poles  of  the  same  sign  turned  towards  each  other. 
If  the  oscillator  held  in  the  hand  be  turned  around,  so  that  the 
two  bodies  are  in  the  same  phase,  the  result  will  be  repulsion  (Fig. 
15,  b),  while  the  corresponding  magnets,  which  have  opposite 
poles  facing  each  other,  will  attract  each  other.  Finally,  the  oscil- 
lator may  be  brought  into  the  position  (Fig.  15,  e)  in  which  it  oscil- 
lates in   the   direction    of  the  prolonged  arm    of  the  oscillation- 


FiG.  15. 

balance.  Then  we  shall  again  get  the  small  lateral  displacement, 
which  brings  the  attracting  poles  of  the  two  oscillating  bodies  near 
each  other. 

The  oscillator  in  the  balance  may  now  be  turned  around  90°,  so 
that  its  oscillation  is  at  right  angles  to  the  direction  in  which  it  is 
free  to  move.  If  both  bodies  oscillate  normally  to  the  line  join- 
ing them,  we  get  attraction  when  the  bodies  oscillate  in  the  same 
phase  (Fig.  15,  c),  and  repulsion  when  they  oscillate  in  the  oppo- 
site phase  (Fig.  15,  d).  This  corresponds  to  the  attraction  and 
repulsion  between  parallel  magnets,  except  that  the  direction  of  the 


40  FIELDS    OF    FORCE. 

force  is,  as  usual,  the  reverse,  the  magnets  repelling  in  the  case  of 
similar,  and  attracting  in  the  case  of  opposite  parallelism.  If, 
finally,  we  place  the  oscillator  in  the  prolonged  arm  of  the  bal- 
ance with  its  axis  of  oscillation  perpendicular  to  this  arm  (Fig. 
15,  /'),  we  again  get  the  small  lateral  displacement  described 
above,  exactly  as  with  magnets  in  the  corresponding  positions,  but 
in  the  opposite  direction. 

AVe  have  considered  here  only  the  most  important  positions  of 
the  two  oscillating  bodies  and  of  the  corresponding  magnets.  Be- 
tween these  principal  positions,  which  are  all  distinguished  by  cer- 
tain properties  of  symmetry,  there  is  an  infinite  number  of  dis- 
symmetric positions.  In  all  of  them  it  is  easily  shown  that  the 
force  inversely  corresponds  to  that  between  two  magnets  in  the 
corresponding  positions. 

11.  Rotations  of  the  Oscillating  Body. — We  have  con^^idered 
hitherto  only  the  resultant  force  on  the  oscillating  body.  But  in 
general  the  two  forces  acting  at  the  two  extreme  positions  also  form 
a  couple,  like  the  two  forces  acting  on  the  two  poles  of  a  mag- 
net. The  first  eifect  of 'this  couple  is  to  rotate  the  axis  of  oscil- 
lation of  the  l)ody.  But  if  this  axis  of  oscillation  has  a  fixed 
direction  in  the  body,  as  is  the  case  in  our  experiments,  the  body, 
will   be  obliged  to  follow  the   rotation  of  the  axis  of  oscillation. 

To  show  the  effect  of  this  couple  experimentally  the  oscillator 
may  be  placed  directly  in  the  cylinder  c  (Fig.  11)  of  the  pulsa- 
tion-balance. It  is  then  free  to  turn  about  a  vertical  axis  passing 
through  the  })ivot.  If  a  pulsating  body  be  brought  into  the  neigh- 
borhood of  this  oscillating  body,  it  immediately  turns  about  its 
axis  until  the  position  of  greatest  attraction  is  reached,  and  as  a 
consequence  of  its  inertia  it  will  generally  go  through  a  series  of 
oscillations  about  this  position  of  equilibrium.  If  the  phase  of 
the  pulsations  be  changed,  the  oscillating  body  will  turn  around 
until  its  other  pole  comes  as  near  as  possible  to  the  pulsating  body. 
Apart  from  the  direction  of  the  force,  the  phenomena  is  exactly 
the  analogue  of  a  suspended  needle  acted  upon  by  a  magnetic  pole. 

The  pulsating  ImxIv  may  now  be  replaced  by  an  oscillating  body. 


INVESTIGATION    OF    DYNAMTCAI.    PROPERTIES.  41 

Except  for  the  direction  of  the  force,  we  shall  get  rotations  corres- 
ponding to  those  of  a  compass  needle  nnder  the  influence  of  a 
magnet.  The  position  of  equilibrium  is  always  the  position  of 
greatest  attraction  (Fig.  15,  «,  c),  the  position  of  greatest  repul- 
sion being  a  position  of  unstable  equilibrium.  If  the  fixed  oscil- 
lating body  oscillates  parallel  to  the  line  drawn  from  its  center  to 
that  of  the  body  in  the  balance,  the  position  of  stable  equilibrium 
will  be  that  indicated  in  Fig.  16,  h,  and  if  it  oscillates  at  right 
angles  to  this  line,  it  will  be  the  position  indicated  in  Fig.  16,  d, 
while  the  intermediate  dissymmetric  positions  of  the  fixed  oscil- 
lator give  intermediate  dissymmetric  positions  of  equilibrium  of 
the  movable  oscillating  body.     It  is  easily  verified  that  the  posi- 


■'  (us': 


c  d 


Fig.  16. 


tions  of  equilibrium  are  exactly  the  same  as  for  the  case  of  two 
magnets,  except  for  the  difference  which  is  a  consequence  of  the 
opposite  pole-law  ;  the  position  of  stable  equilibrium  in  the  mag- 
netic experiment  is  a  position  of  unstable  equilibrium  in  the 
hydrodynamic  experiment,  and  vice  versa. 

12.  Forces  Analogous  to  Those  of  Temporary  Magnetism. — 
We  have  already  considered  the  forces  between  bodies  wdiich  are 
themselves  the  primary  cause  of  the  field,  namely  the  bodies 
which  have  forced  pulsations  or  oscillations.  But,  as  we  have 
shown,  bodies  which  are  themselves  neutral  but  which  have 
another  density  than  that  of  the  fluid  also  exert  a  marked  influ- 
ence upon  the  configuration  of  the  field,  exactly  analogous  to  that 
exerted  by  bodies  of  different  inductivity  upon  the  configuration 
of  the  electric  field.  This  action  of  the  bodies  upon  the  geomet- 
rical configuration  of  the  field  is,  in  the  case  of  electricity  or  mag- 
6 


42 


FIELDS    OF    FORCE. 


netism,  accompanied  by  a  mechanical  force  exerted  by  the  field 
u}3on  the  bodies.  AVe  shall  see  how  it  is  in  this  respect  in  the 
hydrodyuamic  field. 

As  we  concluded  from  the  principle  of  kinetic  buoyancy,  a  body 
which  is  lighter  than  the  water  is  brought  into  oscillation  with 
greater  amplitudes  than  those  of  the  water ;  a  body  of  the  same 
density  as  the  water  will  be  brought  into  oscillation  with  exactly 
the  same  amplitude  as  the  water ;  and  a  body  which  has  greater 
density  than  the  water  will  be  brought  into  oscillation  with  smaller 
amplitudes  than  those  of  the  water.  From  this  we  conclude 
that  during  the  oscillations  the  body  of  the  same  density  as  the 
water  will  be  always  contained  in  the  same  mass  of  water.  But 
both  the  light  and  the  heavy  body  will  in  the  two  extreme  posi- 
tions be  in  different  masses  of  water,  and  if  these  have  not  exactly 


Fjo.  17. 

tiic  same  motion,  it  will  be  subject  in  these  two  positions  to  kinetic 
buoyancies  not  exactly  e(iual  and  not  exactly  opposite  in  direc- 
tion. The  motion  cannot  therefore  be  strictly  periodic.  As  a 
consequence  of  a  feeble  dissymmetry  there  will  be  superposed 
ujx)n  the  oscillation  a  progressive  motion. 

That  the  average  force  which  produces  this  progressive  motion 
is  strictly  analdgous  to  the  force  de])euding  upon  induced  rqagnetism 
or  eloftriliciitioM  by  iuHuence,  is  easily  seen.  As  we  have  already 
shown  in  the  ])receding  lecture,  the  induced  oscillations  correspond 
exactly  to  the  induced  states  of  polarization  in  the  electric  or  the 
magnetic  field.      Further,  the  forces  acting  in  the  two  extreme  posi- 


INVESTIGATION    OF    DYNAMICAL    PROPERTIES. 


43 


tions  of  oscillation  are  in  the  same  relation  to  the  geometry  of  the 
field  as  the  forces  acting  on  the  poles  of  the  induced  magnets ;  they 
are  directed  along  the  lines  of  force  of  the  field,  and  vary  in  inten- 
sity from  place  to  place  according  to  the  same  law  in  the  two  kinds 
of  fields,  except  that  the  direction  of  the  force  is  always  opposite  in 
the  two  cases.  Fig.  17,  a  shows  these  forces  in  the  two  extreme 
positions  of  a  light  body,  which  oscillates  with  greater  amplitudes 
than  the  fluid,  and  Fig.  1  7,  b  shows  the  corresponding  forces  acting 
on  the  two  poles  of  a  magnetic  body.  Therefore,  in  the  hydro- 
dynamic  field,  the  light  body  will  be  subject  to  a  force  oppositely 
equivalent  to  that  to  which  the  magnetic  body  in  the  corresponding 
magnetic  field  is  subject.  Fig.  18,  a  shows  the  forces  acting  on 
the  heavy  body  in  its  two  extreme  positions,  the  oscillations  repre- 
sented in  the  figure  being  those  which  it  makes  relatively  to  the 


Fig.  18. 

fluid,  which  is  the  oscillation  which  brings  it  into  water  masses 
with  different  motions.  Fig.  18,  6  shows  the  corresponding  forces 
acting  on  the  poles  of  an  induced  magnet  of  diaraagnetic  polarity. 
And,  as  is  evident  at  once  from  the  similarity  of  these  figures, 
the  heavy  body  in  the  hydrodynaraic  field  will  be  acted  upon  by  a 
force  which  oppositely  corresponds  to  the  force  to  which  a  diamag- 
netic  body  is  subject  in  the  magnetic  field. 

The  well  known  laws  for  the  motion  of  magnetic  and  diamag- 
netic  bodies  in  the  magnetic  field  can,  therefore,  be  transferred  at 
once  to  the  motion  of  the  light  and  heavy  bodies  in  the  hydro- 
dynamic   field.     The   most  convenient  of  these  laws  is  that  of 


44  FIELDS    OF    FORCE. 

Faraday,  which  connects  the  force  with  the  absolute  intensity, 
or  to  the  energy,  of  the  field.  Remembering  the  reversed  direc- 
tion of  the  force,  we  conclude  that : 

The  lif/ht  body  rvUl  move  in  the  direction  of  decreasing,  the  heavy 
body  in  the  direction  of  increasing  energy  of  the  field. 

13.  Attraction  and  Repulsion  of  Light  and  Heavy  Bodies  by  a 
Pulsating  or  an  Oscillating  Body.  —  If  the  field  be  produced  by 
only  one  pulsating  or  one  oscillating  body,  the  result  is  very 
simple.  For  the  energy  of  the  field  has  its  maximum  at  the  sur- 
face of  the  pulsating  or  oscillating  body,  and  will  always  decrease 
witli  increasing  distance.  Therefore,  the  light  body  will  be  re- 
pelled, and  tlie  heavy  body  attracted  by  the  pulsating  or  the  oscil- 
lating body. 

To  make  this  experiment  we  suspend  in  the  water  from  a  cork 
floating  on  the  surface  a  heavy  body,  say  a  ball  of  sealing  wax. 
In  a  similar  manner  we  may  attach  a  light  body  by  a  thread  to  a 
sinker,  which  either  slides  with  a  minimum  pressure  along  the 
bottom  of  the  tank,  or  which  is  itself  held  up  in  a  suitable  manner 
by  corks  floating  on  the  surface.  It  is  important  to  remark-  that 
the  light  body  should  never  be  fastened  directly  to  the  sinker,  but 
by  a  thread  of  sufficient  length  to  insure  freedom  of  motion. 

On  bringing  a  pulsator  up  to  the  light  body,  it  is  seen  at  once 
to  be  rej)clled.  If  one  is  sufficiently  near,  the  small  induced 
oscillations  of  the  light  body  may  also  be  observed.  If  the  pul- 
sating body  be  brought  near  the  heavy  body,  an  attraction  of  simi- 
lar intensity  is  observed.  In  both  cases  it  is  seen  that  the  force 
de<;reases  much  more  rapidly  with  the  distance  than  in  all  the 
previous  experiments,  the  force  decreasing,  as  is  easily  proved, 
as  the  inverse  fifth  power  of  the  distance,  which  is  the  same  law  of 
distance  found  for  the  action  between  a  magnetic  pole  and  a  piece 
of  iron. 

U  for  tlie  ]>ulsating  body  we  substitute  an  oscillating  body,  the 
same  attractions  and  repulsions  are  observed.  Both  poles  of  the 
osci Hating  body  exert  exactly  the  same  attraction  on  the  heavy 
body,  and  exactly  the  same  repulsion  on  the  light  body,  and  even 


IXVESTIGATION    OF    DYNAMICAL    PROPERTIES.  45 

the  equatorial  parts  of  the  oscillating  body  exert  the  same  attract- 
ing or  repelling  force,  though  to  a  less  degree.  As  is  easily  seen, 
we  have  also  in  this  respect  a  perfect  analogy  to  the  action  of  a  mag- 
net on  a  piece  of  soft  iron,  or  on  a  piece  of  bismuth. 

14.  Simultaneous  Permanent  and  Temporary  Force.  —  As  the 
force  depending  upon  the  induced  pulsations,  oscillations,  or  mag- 
netizations, decreases  more  rapidly  with  increasing  distance  than 
the  force  depending  upon  the  permanent  pulsations,  oscillations,  or 
magnetizations,  very  striking  effects  may  be  obtained  as  the  result 
of  the  simultaneous  action  of  forces  of  both  kinds.  And  these 
effects  offer  good  evidence  of  the  true  nature  of  the  analogy. 

For  one  of  the  simplest  magnetic  experiments  we  can  take  a 
strong  and  a  weak  magnet,  one  of  which  is  freely  suspended.  At  a 
distance,  the  poles  of  the  same  sign  will  repel  each  other.  But 
if  they  be  brought  sufficiently  near  each  other,  there  will  appear 
an  attraction  depending  upon  the  induced  magnetization.  This 
induced  magnetization  is  of  a  strictly  temporary  nature,  for  the 
experiment  may  be  repeated  any  number  of  times. 

We  can  repeat  the  experiment  using  the  pulsation -balance  and 
two  pulsators,  giving  them  opposite  pulsations  but  with  very  dif- 
ferent amplitudes.  At  a  distance,  they  will  repel  each  other,  but 
if  they  be  brought  sufficiently  near  together,  they  will  attract.  It 
is  the  attraction  of  one  body,  considered  as  a  neutral  body  heavier 
than  the  water,  by  another  which  has  intense  pulsations. 

Many  experiments  of  this  nature,  with  a  force  changing  at  a 
critical  point  from  attraction  to  repulsion,  may  be  made,  all  show- 
ing in  the  most  striking  way  the  analogy  between  the  magnetic 
and  the  hydrodynamic  forces. 

15.  Orientation  of  Cylindrical  Bodie.^. — The  most  common 
method  of  testing  a  body  with  respect  to  magnetism  or  diamagnet- 
ism  is  to  suspend  a  long  narrow  cylindrical  piece  of  the  body  in 
the  neighborhood  of  a  sufficiently  powerful  electromagnet.  The 
cylinder  of  the  magnetic  body  then  takes  the  longitudinal,  and 
the  cylinder  of  the  diamagnetic  body  the  transverse  position. 

The  corresponding  hydrodynamic  experiment  is  easily  made 


46  FIELDS    OF   FORCE. 

The  liglit  cvlinder  is  attached  from  below  and  the  heavy  cylinder 
from  above,  and  on  bringing  near  a  pnlsating  or  an  oscillating 
body,  it  is  seen  at  once  that  the  light  cylinder,  which  corresponds 
to  the  magnetic  body,  takes  the  transverse,  and  the  heavy  cylinder, 
which  corres})onds  to  the  diamagnetic  body,  the  longitudinal  position. 

16.  Xenfral  Bodies  Acted  Upon  by  Two  or  3fore  Pulsating  or 
(hcUUdinr/  Bodies.  —  The  force  exerted  by  two  magnets  on  a  piece 
of  iron  is  generally  not  the  resultant  found  according  to  the  paral- 
lelogram-law from  the  two  forces  which  each  magnet  would  exert 
by  itself  if  the  other  were  removed.  For  the  direction  of  the 
greatest  increase  or  decrease  of  the  energy  in  the  field  due  to  both 
magnets  is  in  general  altogether  different  from  the  ])arallelogram- 
resultant  of  the  two  vectors  which  give  the  direction  of  this  increase 
or  decrease  in  the  fields  of  the  two  magnets  separately.  It  is  there- 
fore not  astonishing  that  we  get  results  which  are  in  the  most 
striking  contrast  to  the  principle  of  the  parallelogram  of  forces, 
considered,  it  must  be  emphasized,  as  a  physical  principle,  not 
merely  as  a  mathematical  ])rinciple ;  /.  e.,  as  a  means  of  the 
abstract  representation  of  one  vector  as  the  sum  of  two  or  more 
other  vectors. 

In  this  way  we  may  meet  with  very  peculiar  phenomena,  which 
have  great  interest  here,  because  they  are  well  suited  to  show 
how  the  analogy  between  hydrodynamic  and  magnetic  phenomena 
goes  even  into  the  most  minute  details.  We  shall  consider  here 
only  the  simplest  instance  of  a  phenomenon  of  this  kind. 

Let  a  piece  of  iron  be  attached  to  a  cork  floating  on  the  surface 
of  the  water.  If  a  magnetic  north  pole  be  placed  in  the  water  a 
little  below  the  surfiice,  the  piece  of  iron  will  be  attracted  to  a 
point  verti(!aliy  above  the  pole.  If  a  south  pole  be  placed  in  the 
same  vertical  symmetrically  above  the  surface,  nothing  peculiar  is 
ol)serve<l;  the  |)iece  of  iron  is  held  in  its  position  of  equilibrium 
more  strongly  than  before.  But  if  the  second  pole  be  a  north  pole, 
the  iron  will  seem  to  be  repelled  from  the  point  where  it  had  pre- 
viously stable  e(juilil)rium.  It  will  move  out  to  some  point  on  a 
circle,  the  diainrtcr  of  which  is  about  j7_  of  the  distance  between  the 


INVESTIGATION    OF    DYNAMICAL    PROPERTIES.  47 

poles.  If  the  same  experiment  were  made  with  a  piece  of  bis- 
muth aud  sufficiently  strong  magnetic  poles,  the  force  would  be 
in  every  case  the  reverse.  It  would  have  unstable  equilibrium  in 
the  central  point  between  two  poles  of  opposite  sign,  and  would 
seem  to  be  repelled  from  this  point.  But  if  the  two  poles  were 
of  the  same  sign,  the  bismuth  would  seem  to  be  attracted  to  the 
point  which  previously  repelled  it,  and  it  would  be  drawn  to  this 
point  from  any  point  within  the  circle  mentioned  above.  On 
the  circle  itself  it  would  have  unstable  equilibrium,  and  outside 
it  would  be  repelled. 

These  peculiar  phenomena  are  at  once  understood  if  we  re- 
member that  the  central  point  between  two  poles  of  the  same 
sign  is  a  neutral  point,  where  the  energy  of  the  field  has  a  mini- 
mum (Fig.  5,  b),  and  that  the  bismuth  must  move  towards  this 
point,  the  iron  from  it. 

To  make  the  corresponding  hydrodynamic  experiment  two  pul- 
sators  may  be  placed  one  vertically  above  the  other,  and  a  light 
body  (Fig.  19,  a)  or  a  heavy  body  (Fig.  19,  b)  brought  between 
them.  Then  if  they  pulsate  in  opposite  phase,  the  light  body  will 
be  repelled  from,  and  the  heavy  body  attracted  to  the  central  point 
between  the  two  pulsating  bodies.  But  if  the  phase  be  changed,  so 
that  the  two  bodies  pulsate  in  the  same  phase,  the  light  body  will 
be  attracted  to  this  central  point  from  all  points  inside  a  circle 
whose  diameter  is  about  Jq  of  the  distance  between  the  pulsating 
bodies.  At  all  points  outside  of  this  circle  it  will  be  repelled.  The 
heavy  body,  on  the  other  hand,  will  be  repelled  from  the  center  to 
some  point  on  the  circle,  but  attracted  from  any  point  outside  the 
circle,  so  that  it  will  be  in  stable  ecpulibrium  on  the  circle. 

17.  Mutudl  Reactions  between  Bodies  with  Induced  Magnetiza- 
tiom  or  with  induced  Oscillations.  —  Besides  the  direct  actions  of 
magnets  on  a  piece  of  soft  iron,  we  have  also  actions  between  any 
two  pieces  of  soft  iron  which  are  acted  upon  by  a  magnet.  This  is 
of  special  interest,  because  it  is  upon  this  that  the  formation  of  the 
representations  of  fields  of  force  in  the  classical  experiment  with 
iron  filings  depends.^    The  iron  filings  lying  in  the  same  line  of 


48 


FIELDS    OF    FORCP:. 


force  acquire  poles  of  opposite  sign  facing  each  other,  and  therefore 
chain  together.  Iron  filings  lying  near  each  other  on  a  line  nor- 
mal to  a  line  of  force  have,  on  the  other  hand,  poles  of  the  same 
sign  facing  each  other,  and  therefore  the  chains  formed  mutually 
repel  each  other,  so  that  they  become  separated  by  distinct  inter- 
vals. It  is  worth  meutiouing  that,  if  the  same  experiment  could 
be  made  with  filings  of  a  diamagnetic  body,  such  as  bismuth,  the 
chain  would  be  formed  in  the  same  way.     For  wdien  the  poles  of 


Fig.  19. 


all   the  filings  are  changed  at  the  same  time,  the  direction  of  the 
forces  between  them  will  be  unchanged. 

Similar  actions  will  be  observed  between  particles  which  take  in- 
duced oscillations  in  the  hydrodynamic  field,  except  for  the  differ- 
ence residting  from  the  direction  of  the  force,  which  is  opposite  in 
every  ca.se.  The  particles,  therefore,  will  chain  together  normally 
to  the  lines  of  flow  in  the  fluid  ;  they  will  arrange  themselves  as 
layers  whici)  follow  the  e(piip()tcntial  surfaces,  and  which,  as  a  con- 
sequence of  mutual  repulsion,  are  separated  from  each  other  by 
empty  spaces.     It  is  indifferent  whether  for  the  experiment  we 


INVESTIGATION    OF    DYNAMICAL    i'ROPERTIES. 


41) 


take  a  light  powder,  which  would  correspond  to  the  iron  filings,  or 
a  heavy  powder,  which  would  correspond  to  the  bismuth  filings. 

For  practical  reasons,  it  is  ])referable  to  use  a  heavy  powder, 
which,  in  order  that  the  experiment  succeed  nicely,  must  be 
fairly  homogeneous.  A  good  powder  may  be  obtained  from  com- 
mon red  lead,  if  both  the  finest  and  the  coarsest  particles  be  re- 
moved by  washing.  This  is  distributed  on  a  glass  plate,  directly 
above  which  is  placed  for  a  few  seconds  a  pulsating  or  an  oscillat- 
ing body  with  very  intense  pulsations  or  oscillations.  The  powder 
immediately  arranges  itself  along  the  expected  curves. 


Fig.  20. 
Fig.  19  gives  the  circles  of  a  section  through  the  spherical  equi- 
potential  surfaces  surrounding  a  pulsating  body,  and  Figs.  20 
and  21  give  the  more  complicated  curves  of  plane  sections  through 
the  equipotential  surfaces  produced  by  two  bodies  pulsating  in  the 
same  and  in  opposite  phase,  respectively.  In  a  similar  way  Fig. 
22  gives  a  section  through  the  system  of  equipotential  surfaces 
around  an  oscillating  body.  As  is  easily  seen,  the  curves  thus 
obtained  are  normal  to  the  lines  of  force  or  of  flow  represented  by 

Figs.  4-8. 
7 


50 


FIELDS    OF    FORCE. 


It  is  worth  remarking  that  the  dynamical  principle  which  ex- 
plains the  formation  of  these  figures  is  the  same  as  that  which 
explains  the  formation  of  Kundt's  dust-figures  in  the  classical 
experiment  for  the  measurement  of  the  velocity  of  sound  in  gases. 
Our  figures  also  show  a  striking  likeness  to  the  ripple  marks  formed 
in  the  sand  along  the  shores  by  the  waves.  And  even  though  the 
dynamical  princi})le  developed  here  does  not  fully  account  for  the 
peculiarities  of  these  ripple  marks,  especially  when  they  have 
great  dimensions,  it  is  certainly  the  principle  which  accounts  for 


Fig.  21. 


the  beginning  of  their  formation.  The  fossil  ripple  marks,  which 
are  well  known  to  the  geologists,  then  prove  that  the  laws  of  hydro- 
dynamic  fiekls  of  force,  which  I  develop  before  you  in  this  lecture, 
were  the  same  in  jirevious  geological  periods  as  they  are  to-day. 

18.  }oHi<r)i  (did  Electric  Currents. —  We  have  obtained  the 
most  complete  analogy  possible  of  hydrodynamic  phenomena  to  the 
phenomena  of  electrostatics  or  of  magnetism,  the  only  difference 
being  that  depending  upon  the  inverse  pole-law. 

Our   investigation    of   the    geometry   of  the    field    showed    us 


INVESTIGATION    OF    DYNAMICAL    PROPERTIES. 


51 


that  we  meet  with  difficulties  if  we  try  to  extend  the  analogy 
beyond  this  point.  The  discovery  of  a  complete  dynamical  analogy 
to  the  phenomena  of  electromagnetism  therefore  necessarily  sup- 
poses a  more  or  less  complete  modification  of  the  views  wiiich  have 
led  us  to  the  discovery  of  the  partial  analogy  already  developed. 
To  prepare  for  a  discovery  of  this  kind  we  can  hardly  do  better 
than  to  discuss  the  other  conditions  which  lead  to  a  partial 
analogy  which  is  related  to  the  analogy  which  we  have  developed, 
although  it  does  not  form  an  immediate  continuation  of  it. 


Ficx.  22. 


As  we  remarked  when  we  discussed  the  geometry  of  the  fields, 
there  is  an  analogy,  discovered  by  v.  Hei.mholtz,  between  the 
magnetic  fields  of  electric  currents  and  hydrodynamic  fields  de- 
pending upon  vortex  motion.  This  geometric  analogy  very  nearly 
forms  a  continuation  of  the  analogy  with  which  we  have  been 
mostly  occupied,  the  only  reason  why  it  cannot  form  a  perfect 
continuation  being  that  the  fluid  vortex  must  always  go  around 
in  the  same  direction,  so  that  a  vortex  of  vibratory  nature  is  im- 
possible.    But  taking  this  analogy  as  it  is,  detached  from  the  pre- 


52  FIELDS   OF    FORCE. 

ceding  analogy,  we  will  examine  whether  in  this  case  also  there 
exists  an  analogy  between  the  dynamics  of  the  two  systems. 

Let  us  first  consider  a  rectilinear  cylindrical  vortex  in  the 
middle  of  a  tank,  which  is  itself  at  rest.  The  motion  of  circu- 
lation around  the  vortex,  which  corresponds  to  the  magnetic  field 
around  the  corresponding  electric  current,  will  be  perfectly  sym- 
metrical. The  distribution  of  the  pressure  will,  therefore,  also 
be  symmetrical,  and  there  will  be  no  resultant  force  against  the 
vortex.  Nor  will  any  such  force  appear  if  a  common  motion  of 
translation  be  communicated  to  the  tank  and  to  the  vortex. 
Otherwise  it  would  be  possible  to  discover  by  an  experiment  of 
this  kind  the  motion  of  the  earth. 

But  now  let  us  suppose  the  motion  of  translation  to  be  given  to 
the  tank  only,  while  the  vortex,  or  a  rotating  rigid  cylinder  sub- 
stituted for  the  vortex,  be  held  still.  There  will  then  be  a  dis- 
symmetry in  the  distribution  of  the  motion  on  the  two  sides  of  the 
rotating  cylinder;  on  one  side,  the  motion  of  translation  will  be 
added  to,  on  the  other  sjde,  subtracted  from  the  motion  of  circulation 
around  the  cylinder.  As  we  have  in  this  case  a  stationary  motion 
depending  upon  a  potential,  there  will  be  in  the  fluid  a  diminutiou 
of  the  j)ressure  proportional  to  the  kinetic  energy  in  the  fluid 
motion,  and  therefore  an  excess  of  pressure  on  the  side  where 
there  is  a  neutralization  of  the  two  motions.  The  cylinder,  there- 
fore, is  driven  transversely  through  the  field,  in  the  direction 
in  which  ihcrc  is  additi(m  of  the  velocities.  This  corresponds 
exactly  to  the  transverse  motion  of  an  electric  current  throup-h  a 
homogeneous  magnetic  field,  but  with  the  same  difference  of  sign 
as  befi)re  ;  the  electric  current  is  driven  iu  the  direction  in  which 
the  field  intensity  due  to  the  current  is  neutralized  by  that  due  to 
the  homogeneous  field. 

The  rectilinear  cylindrical  vortex  which  we  have  considered  may 
now  be  an  element  of  any  vortex.  Therefore  we  may  draw  this 
general  conclusion  ;  the  elements  of  any  vortex  which  is  station- 
ary in  space,  will,  in  the  hydrodynamic  field,  be  subject  to  a  force 
oppositely  corresiwiiding  to  thattc»  which  the  elements  of  the  corre- 


INVESTIGATION    OF    DYNAMICAL    PROPERTIES.  53 

spending  electric  current  are  subject  in  the  corresponding  magnetic 
field.  As  special  consequences,  we  deduce,  for  example,  that  par- 
allel vortices  which  rotate  in  the  same  sense,  and  which  correspond 
thus  to  currents  of  the  same  direction,  will  repel,  while  vortices 
rotating  in  the  opposite  direction  will  attract  each  other. 

As  is  seen  from  this  deduction,  the  condition  that  the  vortices 
should  be  stationary  in  space  is  essential.  If  the  elements  of  the 
vortices  participate  in  the  motion  of  the  surrounding  field,  we  come 
back  to  the  case  where  the  rectilinear  vortex  had  the  same  motion 
as  the  tank,  and  in  this  case  there  was  no  force.  The  analogy 
which  we  have  found  is  therefore  strictly  limited  to  the  case  of 
stationary  electromagnetism.  Thus  for  two  reasons  this  restric- 
tion is  imposed  upon  the  analogy.  As  we  saw  in  the  investiga- 
tion of  the  geometry  of  the  analogy,  the  constancy  of  the  vortices 
makes  liydrodynamic  phenomena  corresponding  to  the  induction 
of  currents  impossible.  Now  we  see  that  the  mechanical  forces 
have  values  analogous  to  those  acting  against  the  electric  cur- 
rents, only  when  the  vortices  which  correspond  to  the  electric 
currents  are  perfectly  stationary  in  space.  The  analogy,  there- 
fore, is  a  limited  one  ;  but  even  in  its  limited  state  it  may  give  us 
suggestions. 

19.  Experiments  with  Rotating  Ci/linders.  —  Simple  cases  of  the 
results  developed  may  easily  be  tested  experimentally.  By  means 
of  turbines  driven  by  air-jets,  we  may  set  metal  cylinders  into  ro- 
tation, which  in  turn  produce  the  required  circulation  of  the  sur- 
rounding water  masses  in  consequence  of  friction.  One  such 
cylinder  may  be  held  in  the  hand  by  means  of  a  suitable  support. 
Another  may  be  introduced  into  the  instrument  previously  used  as 
a  pulsation-  or  oscillation-balance.  It  is  necessary,  hoAvever,  in 
order  to  prevent  the  cylinder  in  the  balance  from  taking  by  itself 
a  translatory  motion  through  the  fluid,  always  to  use  two  oppo- 
sitely rotating  cylinders  which  are  arranged  symmetrically  about 
a  vertical  axis  through  the  pivot  (Fig.  23). 

With  this  instrument,  it  is  easily  shown  that  cylinders  rotating 
in  the  same  direction  repel,  and  that  cylinders  rotating  in  the 
opposite  direction  attract. 


54 


FIELDS    OF    FORCE. 


AVe  have  observed  also  that  a  non-rotating  cylinder  effects  the 
configuration  of  the  hydrodynamic  field,  just  as  a  cylinder    of 


Fig.  23. 


INVESTIGATION    OF    DYNAMICAL    PROPERTIES.  55 

infinite  diamagnetivity  effects  the  magnetic  field  (I,  16).  Even 
this  geometric  analogy  is  accompanied  with  an  inverse  dynamic 
analogy  ;  it  is  easily  seen  that  the  rotating  and  the  resting  cylin- 
der attract  each  other,  just  as  a  wire,  carrying  an  electric  current, 
and  a  diamagnetic  body  repel  each  other. 


III. 

THE   GEOMETRIC   PROPERTIES   OF   ELECTRO- 
MAGNETIC  FIELDS   ACCORDING   TO 
MAXWELL'S   THEORY. 

1.  C.  A.  Bjerknes'  Problems  and  Metlwds. — All  the  phenomena 
investigated  in  the  preceding  lectures  by  elementary  reasoning 
and  experiment  were  found  originally  through  mathematical 
analysis  by  the  late  Professor  C.  A.  Bjerknes.  While  searching 
for  plienomena  of  hydrodynamics  which  should  have  the  appear- 
ance of  actions  at  a  distance,  he  solved  the  problem  of  the  simul- 
taneous motion  of  any  number  of  spherical  bodies  in  a  liquid. 
The  discussion  of  the  solution  led  him  to  results  which  he  verified 
later  by  a  series  of  experiments,  of  which  I  have  shown  you  the 
most  important,  using,  however,  instruments  of  improved  con- 
struction. 

We  apparently  deviate  from  the  historical  method  in  taking  the 
elementary  reasoning  and  experiment  first  and  then  proceeding  to 
the  mathematical  theory.  But  this  deviation  may  in  some  sense 
be  more  ap])arent  than  real.  For  the  phenomena  to  be  examined 
certainly  had  in  the  -mind  of  the  discoverer  the  form  of  ideal 
experiments  long  before  their  final  mathematical  solution  was 
obtained.  And  the  exact  calculations  were,  in  part,  at  least,  pre- 
ceded by  elementary  reasoning,  which  was  not  always  correct 
perhaps  and  of  which  the  greater  part  was  lost  after  the  exact 
mathematical  solution  was  found.  We  may  therefore  have  good 
reason  to  believe  that,  starting  as  we  have  done  with  elementary 
reasoning  and  experiment,  we  have  in  some  sense  restored  the 
original  method  of  the  discoverer,  improved  according  to  our 
present  exact  knowledge  of  the  subject. 

2.  Tlir  Proh/nii  of  Ati(f/o(/ics.  —  Proceeding  now  to  the  mathe- 
matical theory  we  shall  also,  in  one  sense,  deviate  considerably  from 

56 


GEOMETRIC    EQUATIONS    OF    ELECTROMAGNETIC    FIELDS.       57 

the  original  method  followed  by  the  discoverer.  At  the  begin- 
ning the  solution  of  the  problem  of  spheres  was  certainly  the  most 
natural  way  of  submitting  the  vague  anticipations  to  a  rigorous 
test,  for  this  was  the  time  when  the  theory  of  the  action  at  a  dis- 
tance was  predominant,  and  the  discovery  of  the  simplest  and  most 
striking  instances  of  apparent  actions  at  a  distance  was  the  most 
fascinating  result  for  a  man  opposing  this  theory  to  strive  for. 

But  time  has  chano-ed.  The  doctrine  of  action  at  a  distance 
has  been  given  up,  and  it  is  the  aim  of  no  natural  philosopher  to 
oppose  it.  The  time  of  fields  of  force  has  come,  and  it  is  our  aim 
now  to  widen  and  deepen  our  knowledge  of  these  fields.  The 
hydrodynamic  phenomena  discovered  by  C.  A.  Bjerkxes  were 
field  phenomena,  and  their  analogy  to  electrical  phenomena  are 
even  still  more  striking  according  to  our  new  views.  But  the 
change  of  view  also  suggested  a  quite  new  method  of  developing 
the  results,  with  unexpected  facility  and  generality.  Of  course,  if 
there  exists  a  close  analogy  between  hydrodynamic  and  electromag- 
netic fields,  this  analogy  must  be  contained  implicitly  in  the  funda- 
mental equations  of  the  two  kinds  of  fields,  namely  in  the  hydro- 
dynamic  equations  of  motion  on  the  one  hand,  and  in  Maxwell's 
equations  of  the  electromagnetic  field  on  the  other.  And  this  is 
exactly  what  I  am  going  to  show  you,  namely,  that  the  analogy 
may  be  developed  directly  from  these  two  sets  of  equations. 

The  method  thus  indicated  is,  indeed,  perfectly  plain  and  easy. 
There  is  no  difficulty  in  finding  the  properties  of  hydrodynamic 
fields,  and  the  only  real  difficulty  with  which  we  meet  arises  from 
the  imperfection  of  our  knowledge  of  electromagnetic  fields.  To 
lay  the  safest  possible  foundation  for  our  research  we  have  first  to 
analyze  carefully  our  knowledge  of  these  fields.  This  will  be  the 
object  of  the  lectures  of  to-day  and  to-morrow. 

3.  MaxweWs  Theory.  —  Our  knowledge  of  electromagnetic  fields 
is  contained  in  what  is  generally  called  Maxwell's  theory.  This 
theory  does  not  tell  us  what  electromagnetic  fields  are  in  their  true 
nature.  It  is  a  formal  theory,  bearing  upon  two  aspects  of  the 
properties  of  the  fields.     What  are  generally  called  Maxwell's 


58  FIELDS    OF    FORCE. 

equations  give  a  very  full  description  of  the  variation  from  time 
to  time  of  the  geometric  configuration  of  electromagnetic  fields. 
To  this  geometric  theory  is  only  feebly  linked  the  much  less  devel- 
oped theory  of  the  dynamical  properties  of  these  fields. 

INIaxwell's  theory  has  a  central  core,  generally  called  the 
equations  for  the  free  ether,  relating  to  which  there  is  good 
agreement  among  diflFerent  writers.  But  this  agreement  ceases 
when  we  pass  to  the  equations  for  ponderable  bodies  and  for  mov- 
ing media,  and,  as  will  be  seen,  the  full  discussion  of  the  analogy 
will  depend  upon  certain  details  of  the  theory  for  this  general 
case.  Proceeding  to  outline  the  theory,  I  shall  follow  principally 
Oliver  Heaviside,*  whom  I  have  found  to  be  my  safest  guide 
in  this  department  of  physics  for  several  reasons,  of  which  I  will 
emphasize  two ;  that  he  uses  a  perfectly  rational  system  of  units, 
and  that  he  takes  into  consideration  more  fully  than  other  writers 
the  impressed  forces,  which  play  a  great  part,  from  a  certain  point 
of  view  even  the  greatest  part,  in  the  theory  to  be  developed.  But 
instead  of  Heaviside's  I  shall  use  my  own  notation,  chosen  partly 
to  economize  letters,  p.artly  that  analogies  and  contrasts  in  the 
things  shall  be  reflected  in  analogies  and  contrasts  in '  the 
notation. 

In  thus  outlining  Maxwell's  theory  I  wish  to  emphasize  that 
I  do  not  introduce  anything  new.  AVhat  I  introduce  I  have  found 
in  other  authors,  who  were  perfectly  uninfluenced  by  the  search 
for  the  hydrodynamic  analogy.  The  guarantee  for  an  unpreju- 
diced test  of  this  analogy  is,  therefore,  so  far  as  I  can  see,  perfect. 

4.  ImJudivHij.  —  To  a  material  medium  we  attribute  two  con- 
stants, defining  its  specific  properties  in  relation  to  the  two  kinds 
of  fields.  These  two  constants  define,  so  to  speak,  the  readiness 
of  the  medium  to  let  electric  or  magnetic  lines  of  induction  pass, 
and  may  l)c  called  the  electric  inductivity,  a,  and  the  magnetic  in- 
durtivity,  /3, 

We  do  not  know  the  exact  nature  of  the  properties  defined  by 
these  constants.     They  can,  therefore,  not  be  determined  in  abso- 

*  Oliver  IleaviHide,  Electromagnetic  Tlieory.     Vol.  I.     London,  1893. 


GEOMETRIC    EQUATIONS    OF    ELECTRODYNAMIC    FIELDS.       59 

lute  measure.  What  we  can  measure  arc  only  their  ratios  for 
any  two  media 

a  0 

If  a,  and  /3^  be  the  constants  of  the  free  ether,  these  ratios  are 
called  the  specific  inductive  capacities,  electric  and  magnetic 
respectively,  of  the  medium  which  has  the  inductivities  a 
and  /9. 

When  we  consider  thus  the  properties  of  any  medium  in  rela- 
tion to  the  fields  as  defined  by  one  electric  and  by  one  magnetic  con- 
stant only,  we  limit  ourselves  to  the  consideration  of  strictly  iso- 
tropic substances,  which  remain  isotropic  even  when  strained,  as 
is  the  case,  for  instance,  with  liquids.  But  any  degree  of  hetero- 
geneity may  be  allowed.  These  suppositions  give  to  the  fields 
exactly  the  generality  wanted  for  our  purpose. 

5.  Electric  and  3Iagnetic  Vectors.  —  A^^e  will  consider  in  this 
lecture  the  geometric  description  of  electromagnetic  fields.  To  give 
this  description,  a  series  of  special  electric  and  special  magnetic  vec- 
tors has  been  introduced. 

We  believe  that  these  vectors  represent  real  physical  states  exist- 
ing in,  or  real  physical  processes  going  on  in  the  medium  whicli 
is  the  seat  of  the  field.  But  the  nature  of  these  states  or  processes 
is  perfectly  unknown  to  us.  What  still  gives  them,  relatively 
speaking,  a  distinct  physical  meaning  is,  as  we  shall  show  more 
completely  in  the  next  lecture,  that  certain  expressions  formed  by 
the  use  of  these  vectors  represent  quantities,  such  as  energy,  force, 
activity,  etc.,  in  the  common  dynamical  sense  of  these  words. 
These  quantities  can  be  measured  in  absolute  measure.  But  their 
expressions  as  functions  of  the  electric  or  magnetic  vectors  contain 
always  two  quantities  of  unknown  physical  nature.  A\'hcn  once 
the  discovery  of  a  new  law  of  nature  allows  us  to  write  another 
independent  equation  containing  the  same  unknown  quantities,  we 
shall  be  able  to  define  perfectly  the  nature  of  the  electric  and  mag- 
netic vectors,  and  submit  them  to  absolute  measurements  in  the  real 
sense  of  this  expression.      Provisionally,  we  can  only  do  exactly 


()0  FIELDS    OF    FORCE. 

the  same  as  does  the  mathematician  in  problems  where  he  has  more 
unknowns  than  e(|uations,  viz.,  content  ourselves  with  relative 
determinations,  considering  provisionally  one  or  other  of  the  un- 
known ((uantities  as  if  it  were  known.  But  we  retain  the  symbols 
for  the  unknown  quantities  in  all  formulae  bearing  upon  the  pure 
theory  of  electromagnetic  phenomena,  for  this  will  be  the  best 
preparation  for  the  final  solution  of  the  problem. 

This  imperfect  knowledge  is,  of  course,  also  the  reason  why 
our  tlieory  of  electromagnetic  fields  is  split  into  two  different, 
loosely  connected,  parts  ;  first,  the  geometric  theory  of  the  fields, 
where  the  relation  of  the  vectors  to  time  and  space  is  considered 
independently  of  ev^ery  question  of  the  physical  sense  of  the 
vectors ;  and  second,  the  dynamical  theory  of  the  fields,  where 
the  question  of  the  nature  of  the  vectors  is  taken  up,  but  only 
imperfectly  solved. 

G.  C/asslJicatlon  oj  the  Vectors. — The  vectors  introduced  to 
describe  the  fields  may  be  divided  into  classes  differing  from  each 
other  in  their  mathematical  properties,  or  in  the  physical  facts  to 
which  they  rehitc. 

On  the  one  hand,  the  electric  as  well  as  the  magnetic  vectors 
are  divided  in  two  classes,  designated  generally  ixs  forces  and  fluxes. 
As  the  forces  cannot  be  proved  to  have  anything  to  do  with  forces 
in  tlie  classical  dynamical  sense  of  the  word,  a  more  neutral  name 
will  be  preferable.  I  will  therefore  use  the  words  field  intensities 
and  fluxes.  Between  field  intensities  and  fluxes  there  is  this  re- 
lation :  by  the  multiplication  of  a  field  intensity  by  the  induc- 
tivity  of  the  medium  a  corresponding  flux  is  formed. 

Field  intensities  and  fluxes  are  vectors  of  different  physical 
nature.  They  cannot  therefore  be  added  together.  This  is  an 
im|)ortant  remark.  For,  according  to  previous  imperfect  views  of 
the  electromagnetic  problem,  this  distinction  was  not  made,  and 
mu(-Ii  confusion  was  caused  by  the  lumping  together  of  hetero- 
geneous ([uantities.  But  in  the  case  of  electricity,  as  well  as  in 
magnetism,  any  two  field  intensities  may  be  added  together,  like- 
wise any  two  fluxes. 


GEOMETRIC    EQUATIONS    OF    ELECTROMAGNETIC    FIELD.       61 

Taking  now  another  point  of  view,  we  can  divide  the  field  in- 
tensities into  induced,  and  impressed  or  energetic  field  intensities,  and 
the  fluxes  likewise  into  induced,  and  iinpressed  or  energetic  fiuxes. 

Tiie  theory  of  induced  fluxes  and  field  intensities  we  have  to  some 
extent  really  mastered.  Maxwell's  equations  are  the  laws  princi- 
pally obeyed  by  these  vectors.  But  in  order  to  complete  the  sys- 
tem formally,  the  impressed  or  energetic  fluxes  and  field  intensities 
are  introduced.  They  represent  certain  states,  or  processes,  under 
certain  circumstances  existing  in,  or  going  on  in,  the  matter,  and 
which  are  ultimately  the  origin  of  every  electric  or  magnetic  phe- 
nomenon. The  intrinsic  polarization  in  the  permanent  magnet, 
or  in  the  pyroelectric  crystal,  is  therefore  represented  by  vectors 
of  this  class.  They  are  introduced  further  as  auxiliary  vectors 
for  the  representation  of  the  creation  of  electric  energy  by  contact- 
electricity,  in  the  thermopile,  or  in  the  voltaic  battery.  As  the 
existence  or  the  supply  of  electric  or  magnetic  energy  is  related 
ultimately  to  states  or  processes  represented  by  these  vectors,  I 
have  termed  them  energetic  vectors,  a  name  given  originally  by  C. 
A.  Bjerknes  to  the  corresponding  hydrodynamic  vectors. 

From  the  fundamental  vectors  thus  defined  we  may  form  new 
ones  by  the  addition  of  vectors  of  the  same  kind.  Thus  the  ad- 
dition of  the  induced  and  the  energetic  field  intensities  gives  the 
total  or  actual  field  intensities,  and  the  addition  of  the  induced  and 
the  energetic  flux  gives  the  total  or  actual  fluxes.  We  have  thus 
introduced  six  electric  and  six  magnetic  vectors.  But  in  each 
group  of  six  vectors  only  two  are  really  independent  of  each  other, 
and  thus  only  two  are  really  needed  for  the  full  description  of  the 
electric  or  the  magnetic  field.  Which  pair  of  vectors  it  will  be 
convenient  to  choose  as  independent  will  depend  upon  the  nature 
of  the  problem  to  be  treated.  But  a  certain  pair  of  vectors  seems 
in  the  majority  of  cases  to  turn  out  as  the  most  convenient ;  this 
is  the  actual  fl.iLv  and  the  induced  field  inten^^iti/.  These  we  there- 
fore distinguish  beyond  the  others,  calling  them  simply  the  flux, 
and  the  field  intensity,  in  every  case  when  their  qualities  as  actual 
flux  and  induced  field  intensity  need  not  be  specially  emphasized. 


62 


FIELDS    OF    FORCE. 


7.  Notation.  —  It  is  very  convenient  for  our  purpose  to  intro- 
duce such  notation  as  to  make  it  at  once  evident  to  which  class  or 
group  the  vector  belongs.  To  attain  this  I  denote  fluxes  with  capi- 
tals and  field  intensities  with  the  corresponding  small  letters.  On 
the  other  hand,  actual,  induced,  and  energetic  vectors  are  desig- 
nated by  the  subscripts  a,  i,  e,  but  with  the  exception  that  the  letters 
designating  the  flux  and  the  field  intensity,  according  to  the  defini- 
tions above,  are  distinguished  by  the  omission  of  subscripts. 
Finally,  for  the  electric  vectors  I  use  the  first,  and  for  the  magnetic 
vectors  the  second  letter  of  the  Latin  alphabet,  corresponding  to 
the  first  and  second  letter  of  the  Greek  alphabet  introduced  above 
to  represent  the  inductivities. 

The  system  of  notation  is  contained  in  the  following  scheme  : 


(^) 


Electric. 

Magnetic. 

Flux. 

Field  iutensity. 

Flux. 

Field  intensity. 

Actual 

Induced 

Energetic 

A 

A; 

A, 

a.a 

a 

a.e 

B 
B. 

Be 

b 
be 

Electric  inductivity  a. 


Magnetic  inductivity  /?. 


Between  each  group  of  six  vectors  there  are,  according  to  what 
is  stated  above,  four  relations,  namely  : 


(a) 


A  =  A.  +  A^, 
a,  =  a  +  a , 
A.  =  aa, 

K  =  Ota , 


I      '  e' 

b  =  b  +  b , 
B,  =  /3b, 

B,  =  /3b^. 


By  different  eliminations  we  can  of  cour.se  give  different  forms 
to  these  equations  of  connection.  AVlien  we  agree  to  use  the  flux 
and  the  field  intensity  as  the  fundamental  vectors,  we  need  the 
efpiations  of  connection,  especially  if  vectors  of  the  energetic  group 
have    to  be  introduced.     As  we  prefer  generally  in  such   cases 


GEOMETRIC    EQUATIONS    OF    ELECTROMAGNETIC    FIELDS.       63 

to  introduce  the  energetic  flux,  we  shall  usually  have  to  employ 
the  following  form  of  the  equations  of  connection, 

(6)  A  =  aa  +  A  ,         B  =  /3b  +  B^. 

I  am  aware,  of  course,  that  the  multiplication  of  systems  of 
notation,  already  too  numerous,  may  be  objectionable.  But  it  will 
serve  for  my  excuse,  I  hope,  that  suggestive  notations  are  perhaps 
nowhere  of  greater  importance  than  in  researches  of  a  comparative 
nature.  The  question  of  a  system  of  notation,  at  the  same  time 
simple  and  suggestive,  with  reference  to  the  whole  of  theoretical 
physics,  will,  I  think,  necessarily  arise  sooner  or  later. 

8.  Conductivlti/,  Time  of  Relaxation.  —  Besides  their  electric 
and  magnetic  inductivities,  some  or  most  media  have  still  an  in- 
trinsic property,  their  electric  conductivity.  The  constant  best 
suited  to  represent  this  property  in  the  fundamental  equations  is 
the  time  of  relaxation,  introduced  first  by  E.  Cohn.  If  an  elec- 
tric field  in  a  conducting  medium  be  left  to  itself,  its  electric 
energy  will  be  transformed  into  heat,  and  the  electric  field  will 
disappear.  This  may  happen  so  that  the  configuration  of  the  field 
is  left  unaltered  during  this  process  of  relaxation.  Tlie  time  in 
which  the  electric  vector,  during  this  process,  diminishes  to  the 
fraction  l/e  of  its  initial  value  {e  being  the  base  of  the  natural 
logarithms)  is  the  relaxation  time  T.  This  is  a  real  intrinsic  con- 
stant of  the  medium,  measurable  moreover  in  absolute  measure, 
and  therefore  in  theoretical  researches  to  be  preferred  to  the  con- 
ductivity 7,  to  which  it  is  related  by  the  equation 

a 
(«)  ^=  y- 

A  corresponding  magnetic  conductivity  and  time  of  relaxation 
is  not  known.  It  is  convenient,  however,  in  order  to  obtain  a 
perfect  symmetry  of  the  formulae,  to  introduce  symbols  even  for 
these  fictitious  quantities,  say  k  for  magnetic  conductivity  and  T' 
for  the  corresponding  time  of  relaxation 

w  ^'  =  f  • 


(54  FIELDS    OF    FORCE. 

9.  Integral  Form  of  the  Fundamental  Laws.  —  Using  induced 
field  intensity,  actual,  and  energetic  flux,  avb  have  always  to  re- 
member first  the  equations  of  connection 

A  =  aa  +  A^, 
^"^  B  =  ^b  +  B^. 

A  set  of  cross  connections  between  electric  flux  and  magnetic 
field  intensity,  and  vice  versa,  between  magnetic  flux  and  electric 
field  intensity,  is  contained  in  the  two  "  circuital "  laws,  to  use 
Heaviside's  expression.  To  find  the  mathematical  expressions 
of  these  laws  we  consider  a  surface  bordered  by  a  closed  curve.  In 
case  the  medium  is  moving,  the  surface  should  also  move,  follow- 
ing exactly  the  material  particles  with  which  it  coincides  at  the 
beginning.  To  coordinate  the  positive  side  of  the  surface  with 
the  positive  direction  of  circulation  on  the  bordering  curve  we 
utilize  the  positive  screw-rule.  Denoting  by  r  the  radius  vector 
from  a  fixed  origin  to  a  point  of  the  closed  curve,  by  <h  the  vector- 
line  element  of  the  curve,  by  ds  the  vector-surface  element  of  the 
surface,  the  two  circuital  laws  may  be  written 

If*  ■'^=  +  /r  *■''=  =  /"  ■*' 

The  sum  of  the  surface  integrals  of  the  first  equation  is  generally 
called  the  electric  current  through  the  surface,  the  first  represent- 
ing the  displacement-current  in  the  widest  sense  of  this  word,  and 
the  second  the  conduction  current.  In  the  same  way  the  surface- 
integrals  of  the  second  equation  represent  the  magnetic  current, 
the  second  term,  which  represents  the  magnetic  conduction-cur- 
r<'nt,  being  merely  fictitious.  Utilizing  these  expressions,  the  core 
of  our  knowledge  of  the  properties  of  electro-magnetic  fields  in 
relation  to  time  and  space  may  be  expressed  in  the  following 
propositions. 

TJic  r/ectrlc  current  tlirourjh  a  moving  material  surface  equals  the 


CxEOMETRIC    EQUATIONS    OF    ELECTROMAGNETIC    FIELDS.      G5 

'po.ntive  line  integral  of  the  magnetic  field  intensity  round  the  border 
of  the  surface. 

The  magnetic  current  through  a  moving  material  surface  equals 
the  negative  line  integral  of  the  electric  field  intensity  round  the 
border  of  the  surface. 

To  these  equations,  containing  the  two  fundamental  laws,  we 
have  to  add  two  equations  containing  the  definition  of  two  im- 
portant auxiliary  quantities,  the  electric  and  the  magnetic  mass,  or 
equivalently,  the  electric  and  the  magnetic  density.  Calling  E  the 
electric  and  M  the  magnetic  density,  then  the  electric  or  the  mag- 
netic mass  contained  within  a  closed  surtace  is  the  volume  integral 
respectively  of  E  or  M  within  the  surtace.  These  masses  are  de- 
fined as  functions  of  the  fluxes  by  the  equations 

fEdT=  fAds 

fMdr  =  Jb  •  ds, 

dr  being  the  element  of  volume.  Thus  the  mass  within  a  surface 
is  defined  by  the  total  flux  through  the  surface. 

It  must  be  emphasized  that  these  equations  are,  from  our  point 
of  view,  only  equations  of  definition,  not  laws  of  nature.  This 
assertion  is  not  contradicted  by  the  historical  fact  that  the  notion 
of  masses  was  first  introduced,  and  later  the  vectors  defined  by 
use  of  the  masses,  while  we  now  consider  the  vectors  as  the 
fundamental  quantities,  and  define  the  masses  by  the  vectors.  It 
seems  to  be  an  empirical  fact,  however,  that  no  magnetic  mass  ex- 
ists, and  this  assertion  then  contains  a  law  of  nature  to  which  the 
magnetic  flux  is  subject,  and  which  limits  the  generality  of  the 
magnetic  field.  But  for  the  sake  of  analytical  generality  and  the 
advantages  of  a  complete  symmetry,  it  will  be  convenient  to  retain 
the  symbol  M  for  magnetic  density  in  our  formulae. 

To  these  fundamental  equations  a  system  of  supplementary  con- 
ditions is  usually  to  be  added.  Thus,  it  is  generally  considered 
that  the  values  of  each  inductivity,  «  and  fi,  and  the  relaxation 
time  T  are  known  at  all  points  of  the  field.  The  same  supposition 
9 


66  FIELDS    OF    FORCE. 

is  generally  made  for  the  energetic  fluxes,  and  for  the  electric 
and  magnetic  densities.  In  the  case  of  conductors  a  knowledge  of 
the  total  electric  mass  only  for  each  conductor  is  wanted.  This 
sort  of  special  knowledge  is  wanted  only  for  material  bodies,  and 
not  for  the  free  ether.  For  it  is  generally  admitted  that  here  the 
inductivities  have  constant  values,  a,,,  and  '^^,  that  the  relaxation- 
time  is  infinite,  l/T  =  0,  and  that  energetic  vectors  and  electric  or 
magnetic  densities  do  not  exist,  A^  =  B^  =  0,  jE'=  31  =  0.  These 
conditions  very  much  simplify  the  problems. 

10.  Differential  Form  of  the  Fundamental  Equations.  —  From 
the  integral  forms  above  we  can,  by  a  well  known  mathematical 
process,  pass  to  the  differential  form  of  the  same  equations,  and 
thus  arrive  at  the  form  of  the  system  of  electromagnetic  equations 
generally  most  convenient  for  practical  use. 

The  equations  of  connection  of  course  retain  their  form 

A  =  aa  +  A  , 

(«) 

B  =  ySb  +  B^. 

The  equations  expressing  the  two  circuital  laws  may  be  written 
in  the  following  simple  forms, 

(6,)  '  =  "">  "' 

k  =  —  curl  a, 

where  the  auxiliary  quantities  c  and   k  are  the  electric  and   the 

magnetic  current  densities  respectively,  the  full  expressions  for 

which  are 

rA  ,  1 

c  =   .^  +  curl  (A  X  V)  +  (div  A)V  +  ^  A, 


iK) 


k  =  ',.  +  curl  (B  X  V)  +  (div  B)V  -f  ^,B, 


V  being  the  velocity  of  the  moving  medium,  and  djdt  the  local  time 
differentiator,  which  is  related  to  the  individual  time  differentiator 
used  above  by  the  Eulerian  relation 

/7  \  d        d 

<'^=>  *  =  5l  +  ^'^- 


GEOMETRIC    EQUATIONS    OF    ELECTROMAGNETIC    FIELDS. 


67 


The  second  equation  contains  two  terms  which  represent  merely 
fictitious  quantities,  namely,  (div  B)  V,  which  represents  the  mag- 
netic convection-current,  and  \  /  T  B,  which  represents  the  mag- 
netic  conduction  current. 

The  equations  of  definition  of  the  electric  and  magnetic  densities 

finally  take  the  form 

^  E  =  div  A, 

(^)  M=  div  B. 

To  these  fundamental  relations  we  add  the  equations  which  give 
the  special  features  of  the  free  ether,  namely, 

.=.„  J.  =  0,         ^=0,         A.=  0, 

{d)  . 

^=^.  J.-0,        J/=0,         B.  =  0, 

which  are  satisfied  in  all  space  outside  the  bodies. 

1 1 .  SMionary  State.  -  The  principal  feature  of  electromagnetic 
fields,  as  expressed  by  the  equations  above,  is  this  :  every  varia- 
tion in  time  of  an  electric  field  is  connected  with  the  existence  of 
a  magnetic  field  of  a  certain  geometric  quality;  and  vicev^rsa, 
every  variation  in  time  of  a  magnetic  field  is  connected  with  the 
existence  of  an  electric  field  of  a  certain  geometric  quality. 

This  close  cross  connection  of  electric  and  magnetic  phenomena 
is  reduced  to  a  feeble  link  in  the  case  of  stationary  phenomena, 
and  disappears  completely  when  we  pass  to  static  phenomena. 
To  consider  stationary  fields,  that  is,  fields  which  do  not  vary  in 
time,  let  the  medium  be  at  rest,  V  =  0,  and  let  the  v-ectors  A  and 
B  have  values  which  are  at  every  point  of  space  independent  of 
the  time.  The  expressions  (10,6,)  for  the  two  current  densities 
reduce  to 


(a) 


C  =      m  A, 


k=  j,B. 


68  FIELDS   OF    FORCE. 

The  first  of  these  equations  is  the  most  general  expression  for 
Ohm's  law  for  the  conduction-current,  which  is  thus  the  only- 
current  existing  under  stationary  conditions.  The  second  equa- 
tion gives  the  corresponding  law  for  the  fictitious  magnetic  cur- 
rent. The  currents  are  the  quantities  which  connect  the  elec- 
tric fields  with  the  magnetic  fields,  and  vice  versa.  But  utilizing 
the  invariability  of  the  current,  we  can  now  simply  consider  the 
distribution  of  the  currents  in  the  conducting  bodies  as  given, 
and  thus  treat  the  two  stationary  fields  separately,  without  any 
reference  to  each  other. 

Writing  the  equations  of  the  two  stationary  fields,  we  have 

A  =  aa  -f  A  ,  B  =  ySb  -f  B^, 

(i,)  curl  a  =  —  k,  curl  b  =  c, 

div  k=  E,  div  B  =  Jf, 

where  the  current  densities  c  and  k  are  now  among  tjie  quanti- 
ties generally  considered  as  given.  To  these  fundamental  equa- 
tions the  conditions  for  the  free  ether  must  be  added.  The 
condition  that  the  free  ether  has  no  conductivity  implies  now 
that  no  current  whatever  exists  in  it ;  these  conditions  can  be 
written 

^  =  /3o, 

(/.)  ^^=^'  ^^  =  ^'         • 

^=0,  Jf=0, 

k  =  0,  c  =  0, 

for  the  two  fields  respectively. 

Each  of  the  two  systems  of  equations  contains  one  fictitious 
(juantity.  The  equations  for  the  electric  field  contain  the  sta- 
tionary magnetic  current  density  k,  and  the  equations  for  the  elec- 
tric field  contain  the  density  of  magnetism  M,  both  of  which  are 
fictitious. 

12.  static  Slate.— jr,  in  the  equations  for  stationary  fields,  we 
suppose  the  current  density  to    be  everywhere  nil,   we  get    the 


GEOMETRIC    EQUATIONS    OF    ELECTROMAGNETIC    FIELDS.       G9 

equations  for  static  fields, 


(«l) 

A  =  aa  +  A^, 
curl  a  =  0, 
div  A  =  B, 

B  =  ^b  +  B^, 
curl  b  =  0, 
div  B  =  M, 

with 

the  conditions  for  the  free  ether, 

a  =  a^, 

^  =  /3o, 

{\) 

A=0, 

Jf=0. 

These  static  fields  exist  independently  of  each  other,  the  links 
which,  in  the  general  case,  connect  the  one  kind  of  field  to  the 
other,  namely,  the  currents,  being  nil. 

13.  The  Energy  Integral.  — A  research  relating  to  the  com- 
pleteness of  the  description  which  the  preceding  equations  give  of 
the  geometry  of  the  fields  will  be  of  fundamental  importance  in 
the  search  for  the  analogy  of  these  fields  to  other  fields.  As  an 
introduction  to  this  research,  we  will  examine  from  an  analytical 
point  of  view  an  integral,  the  physical  significance  of  which  will 
occupy  us  in  the  next  lecture,  namely,  the  integral  expressing  the 
electric  or  the  magnetic  energy  of  the  field. 

The  expression  for  the  electric  energy  can  always  be  written 

(a)  ^  =  */A-a//T, 

where  the  integration  is  extended  to  all  space.     Now  in  the  case 
of  perfect  isotropy  the  actual  field  intensity  is  related  to  the  flux 
simply  by  the  relation 
(6)  A  =  aa ,, 

and,  therefore,  we  have  the  equivalent  expressions  for  the  energy 


70  FIELDS    OF    FORCE. 

Now  let  us  write  the  vector-factor  A,  of  the  scalar  product,  iu  the 

form 

A  =  —  aS7(f)  +  curl  G, 

expressing  it  thus  by  a  scalar  potential  cf)  and  a  vector  potential 
G,  as  is  possible  with  any  vector.  The  integral  may  then  be 
written 

4>  =  —  hfA  ■  V(f)dT  +  J*ia,^  •  curl  Gch. 

To  avoid  circumlocution  we  shall  suppose  that  there  exists  in 
the  field  no  real  discontinuity,  every  apparent  surface  of  discon- 
tinuity being  in  reality  an  extremely  thi-n  sheet,  in  which  the 
scalars  or  the  vectors  of  the  field  chauo-e  their  values  at  an  exceed- 
ingly  rapid  rate,  but  always  continuously.  Further,  we  suppose 
that  the  field  disappears  at  infinity.  Both  integrals  can  be  trans- 
formed then  according  to  well  known  formulae,  giving  for  the  en- 
ergy the  new  expression 

(d)  ^  =  hf<j)  div  Adr  +  i  Jg  •  curl  a //t. 

Now  div  A  is  the  density  of  true  electrification,  which  e.xists 
only  in  material  bodies.  It  will  be  sufficient,  therefore,  to  apply 
the  first  integral  to  material  bodies  only,  and  not  to  the  surround- 
ing ether.  If  we  split  the  actual  field  intensity  a^  into  its  induced 
and  energetic  parts,  we  get 

curl  a,  =  curl  a  +  curl  a.. 

Here,  according  to  the  fundamental  equations,  —  curl  a  repre- 
sents the  magnetic  current  k.  By  analogy,  —  curl  a^  can  also  be 
said  to  represent  a  magnetic  current  k  .  By  this  current  the  in- 
trinsic polarization,  say  in  a  turmaline-crystal,  can  be  represented, 
in  the  same  way  as  the  intrinsic  magnetization  can  be  represented, 
according  to  Ampere's  theory,  by  a  distribution  of  electric  cur- 
rents. 

Now  in  the  case  of  a  stationary  field  the  current  —  curl  a  can 
only  exist  in  material  bodies,  not  in  the  surrounding  ether.  And 
the  current— curl  a^,  or  the  vector  a^=l/aA^from  which  it  is 


GEOMETRIC    EQUATIONS    OF    ELECTROMAGNETIC    FIELDS.      71 

derived,  never  exists  outside  material  bodies  (10,  d).  Therefore, 
in  the  case  of  a  stationary  field  it  will  be  sufficient  to  apply  the 
second  integral  in  (d)  to  material  bodies  only. 

From  the  symmetry  of  the  two  sets  of  equations  it  is  seen  at 
once  that  the  integral  expressing  the  energy  of  the  magnetic  field, 
namely, 

^  =  ^  Jb  •  h/h, 

can  be  transformed  to  a  form  corresponding  to  (d),  namely, 

^  =  ij-v/r  div  Bdr  +  i/H  •  curl  hdr, 

involving  thus  the  true  density  of  magnetism,  div  B,  and  the 
electric  current,  curl  B^,  which  is  made  up  of  the  true  electric  cur- 
rent, curl  b,  and  the  fictitious  current,  curl  b^,  by  which,  accord- 
ing to  Ampere's  theory,  the  intrinsic  magnetization  can  be  repre- 
sented. Now  under  stationary  conditions  the  true  current,  curl 
a,  only  exists  in  material  bodies,  not  in  the  ether,  and  the  quantities 
div  B  and  curl  b^  never  exist  except  in  material  bodies. 

We  can  therefore  assert  that  the  energy  of  the  stationary  field, 
whether  it  be  electric  or  magnetic,  can  be  expressed  by  integrals 
which  apply  to  material  bodies  only,  not  to  the  surrounding  ether. 

15.    Conditions  for  the  Vanishing  of  tlie  Stationary  Field. — Let 

us  consider  now  the   stationary  electric   field  in  the  case  where 

there  exists  no  true  electrification,  div  A  =  0,  no  energetic  field 

intensity,  a^  =  0,  and  no  true  current,  curl  a  =  0.     We  shall  then 

have 

div  A  =  0,  curl  a^  =  0, 

and  under  these  circumstances  the  energy  of  the  field  disappears 
completely,  as  is  seen  from  the  expression  (13,  d).  But  accord- 
ing to  the  expression  (13,  c)  for  the  same  energy,  which  is  the 
sum  of  only  positive  elements,  the  flux  A  must  disappear  in  every 
part  of  the  field.  But  when  both  the  flux  A  and  the  energetic 
field  intensity  disappear,  it  is  seen  from  the  equations  of  connec- 
tion that  the  induced  field  intensity  will  also  disappear,  and  there 


72  FIELDS    OF    FORCE. 

will  exist  no  electric  field  at  all.  lu  the  case  of  the  magnetic 
field  perfisctly  parallel  conclusions  can  be  drawn.     Thus  : 

If  there  exists  no  true  electrification,  no  energetic  fiiix,  and  no 
magnetic  current,  there  ivill  exist  no  stationary  electric  field. 

If  there  exists  no  true  magnetism,  no  energetic  flux,  and  no  elec- 
tric current,  there  will  exist  no  stationary  magnetic  field. 

10.  Unique  Deter  mi  nateness  of  the  Stationary  Field. — From  this 
result  a  new  one  can  be  drawn  at  once.  Let  us  consider  two 
fields,  represented  by  the  vectors  A,  a,  and  A',  a',  both  subject  to 
the  condition  of  having  the  same  distribution  of  the  energetic 
flux  A^,  of  the  magnetic  current  k,  and  of  the  true  electrification 
E.     The  equations  of  the  two  fields  will  then  be 

A  =  Ota  +  A^,  A'  =  aa'  +  A^, 

curl  a  =  —  k,  curl  a'  =  —  k, 

div  k  =  E,  div  A'  =  E. 

T>et  us  consider  next  the  field  represented  by  the  difference  of  the 
vectors  of  the  two  fields,  i.  e.,  the  field 

A"  ==  A'  -  A, 
a"  =  a'  —  a. 

As  is  seen  at  once,  this  field  will  be  subject  to  the  conditions 

A"  =  aa", 
curl  a"  =  0, 
div  A"  =  0. 

It  will  thus  be  a  field  having  no  energetic  flux,  no  magnetic  cur- 
rent, and  no  true  electrification,  and  it  will  disappear  completely 
according  to  the  result  above.  Thus  the  fields  A,  a,  and  A',  a', 
cannot  differ  from  each  other. 

Perfectly  ])aralU'l  developments  can  be  given  for  the  magnetic 
field,  aud  we  arrive;  thus  at  the  following  parallel  results  : 


GEOMETRIC    EQUATIONS   OK    ELECrKOMAONBTK;    FIELDS,     73 

According  to  our  system  of  equations,  Ihe  Miomry  declrk  field 
;.,  nn^nch  deU-nninM  by  the  didribuHon  of  true  eleeinfieal^on,  of 
eneJio  Lrh  fl«x,  and  of  rnagneUc  curraU ;  and  (Ae  «ta(.on«,-y 
„umeth  field  is  nniquehi  determined  by  the  dMr,bufiO»  of  true 
Jejnellm,  of  enerejeile  .uujnefu;  fl,„;  and  of  eleMe  cun-ent. 

Tliese  the'orems  show  tlie  amount  of  knowledge  of  the  geometry 
of  the  stationary  fields  which  is  laid  down  in  the  equations  11,  6. 
They  contain  iu  the  most  condensed  form  possible  onr  whole  knowl- 
edjof  this  geometry.  And  the  importance  of  these  theorems 
for  our  purpose  is  perfectly  clear:  if  we  succeed  later  ,n  represent- 
g  the  hydrodynamic  field  by  a  similar  system  of  equations,  there 
will,  u„d  r  similar  conditions,  be  no  chance  for  ditference  in  the 
geometric  properties  of  the  hydrodynamic  field,  and  the  stationary 

electric  or  magnetic  field.  ,     ,     t     i 

But  before  we  proceed  t«  the  investigation  of  the  hydrodynamic 
field  we  have  to  consider  the  dynamic  properties  of  the  electric  and 
the  magnetic  field. 


10 


IV. 

THE    DYNAMIC    PROPERTIES    OF     ELECTROMAG- 
NETIC   FIELDS    ACCORDING    TO 
MAXWELL'S   THEORY. 

1.  Electvio  and  Magnct'iG  Etiergi/.  —  The  Maxwell  equations 
give,  as  I  have  emphasized,  only  a  geometric  theory,  bearing  upon 
tlie  distribution  in  space  of  a  series  of  vectors  whose  physical 
meaning  is  perfectly  unknown  to  us.  To  give  this  theory  a  phys- 
ical content  an  additional  knowledge  is  wanted,  and  this  is  afforded 
by  our  experience  relating  to  the  transformations  of  energy  in  the 
electromagnetic  field. 

The  safest  way,  in  our  present  state  of  knowledge,  of  establish- 
ing this  dynamical  theory  of  the  electromagnetic  field,  seems  to 
be  this ;  start  with  the  expression  which  is  believed  to  represent 
the  energy  of  the  electric  and  of  the  magnetic  field,  and  bring 
into  api^lication  the  universal  principle  of  the  conservation  of 
energy. 

The  general  feature  of  the  method  to  be  used  is  thus  perfectly 
clear;  nevertheless,  the  details  will  be  open  to  discussion.  First 
of  all,  there  is  no  perfect  accordance  between  the  different  writers 
with  regard  to  the  true  expression  of  the  energy  of  the  fields. 
All  authors  agree  that  it  is  a  volume  integral  in  which  the  func- 
tion to  be  integrated  is  the  half  scalar  product  of  a  flux  and  a  field 
intensity.  ]3nt  opinions  seem  to  differ  as  to  whetlier  it  should  be 
the  actual  fluxes  and  field  intensities  or  only  the  induced  ones.  Fol- 
lowing Heavtside,  I  suppose  that  the  «c^wa^  fluxes  and  field  inten- 
sities are  the  proper  vectors  for  expressing  the  energy,  and  thus 
write  the  expression  for  the  total  energy  of  the  electromagnetic  field 

cl>  +  >p  =  J  1 A  •  slJt  +  /  iB  ■  hdr. 

Here,  the  first  integral  gives  the  amount  of  Hie  electric,  and  the 

74 


DYNAMIC    EQUATIONS    OF    ELECTROMAGNETIC    FIELDS.         75 

second  the  amount  of  the  magnetic  energy,  the  integrations  being 
extended  over  the  whole  field. 

2.  Localization  and  Continuity  of  Energy.  —  Starting  with  this 
expression  for  the  energy  of  the  field  and  bringing  into  application 
the  principle  of  the  conservation  of  energy,  we  can  of  course  de- 
duce only  results  strictly  in  accordance  with  the  experience  which 
led  us  to  tliis  form  of  expression  for  the  energy.  We  are  able 
then  to  derive  the  amount  of  mechanical  work  done,  and  conse- 
quently the  forces  doing  it,  for  the  case  when  the  different  bodies 
in  the  field  are  displaced  relatively  to  each  other. 

But  for  the  sake  of  the  problem  before  us,  it  is  very  desirable 
to  go  a  step  further,  to  determine  not  only  the  resultant  forces 
acting  against  the  bodies  as  a  whole,  but  also  the  system  of  ele- 
mentary forces,  which  act  upon  the  elements  of  volume  of  the 
bodies,  and  of  which  the  resultant  forces  are  composed.  Of  these 
elementary  forces  we  have  only  a  very  limited  experimental 
knowledge,  and  to  derive  them,  additional  knowledge  is  needed, 
which  is  not  contained  in  the  mere  statements  of  the  form  of  the 
energy  integral  and  of  the  principle  of  the  conservation  of  energy. 
We  do  not  possess  this  in  universally  accepted  form,  but  we  admit 
as  working  hypotheses  the  following  two  principles  : 

First,  we  suppose  that  it  is  allowable  to  speak  not  only  of 
amounts  of  energy,  but  also  of  a  distribution  of  energy  in  space. 
That  this  should  be  so  is,  a  priori,  not  at  all  clear.  The  uni- 
versal principle  of  the  conservation  of  energy  relates  only  to 
amounts  of  energy.  And  in  the  model  science  relating  to  energy, 
abstract  dynamics,  the  notion  of  a  certain  distribution  of  en- 
ergy in  space  seems  to  be  often  of  rather  questionable  clearness 
and  utility.  But  still  it  may  have  a  more  or  less  limited  useful- 
ness. Assuming  this,  we  admit  as  a  working  hypothesis,  that 
the  energy  integral  not  only  gives  the  total  amount  of  electric 
and  magnetic  energy,  but  also  the  distribution  of  this  energy 
in  space,  the  amount  of  energy  per  unit  volume  in  the  field  being 

lA  a  -f-  IB  ■  b  . 


76  FIELDS    OF    FORCE. 

To  this  princii)le  of  the  localization  of  energy  we  add  the  second, 
the  principle  of  the  continuity  of  energy,  which  is  this  :  energy  can- 
not enter  a  space  Avithout  passing  through  the  surface  surrounding 
this  space.  This  principle  forces  us  to  admit  a  more  or  less  de- 
termined motion  of  the  energy,  which  in  connection  with  the  trans- 
formations of  the  energy  regulates  the  distribution  of  the  electro- 
magnetic energy  in  space.  To  this  principle  we  may  make  sim- 
ilar objections  as  to  the  previous  one.  The  idea  of  a  determinate 
motion  of  the  energy  does  not  in  abstract  dynamics  seem  to  be 
always  very  clear  or  useful,  even  though  it  may  seem  to  have 
in  this  branch  of  physics  also  a  certain  limited  meaning.  And 
even  though  considerable  doubt  may  fall  upon  these  two  supposi- 
tions considered  as  universal  principles,  no  deciding  argument  can 
be  given  at  present  against  their  use  to  a  limited  extent  as  work- 
ing hypotheses. 

3.  Electric  and  Magnetic  Activity.  —  To  these  abstract  and 
general  principles  we  have  to  add  definite  suppositions  suggested 
more  or  less  by  experiment.  The  first  is  this  :  the  rate  at  which 
the  electric  or  magnetic  energy  is  created  by  the  foreign  sources 
of  energy  is  given  per  unit  volume  by  the  scalar  product  of  the 
energetic  field  intensity  into  the  corresponding  current.  This 
princij)le  was  originally  suggested  by  the  observation  that  the  rate 
of  doing  work  by  the  voltaic  battery  was  the  product  of  its  in- 
trinsic electromotive  force  and  the  current  produced  by  it.  And 
it  is  generalized  by  inductive  reasoning  so  that  it  is  made  to  in- 
clude every  impressed  or  energetic  force  and  field  intensity,  every 
furrent,  electric  or  magnetic,  conduction  current,  or  displacement 
current. 

Starting  thus  with  Maxwell's  equations  for  the  general  case 
of  a  moving  medium 

c  =  curl  b, 

k  =  —  curl  a, 

we  can  at  once  find  the  rate  at  which  energy  is  supplied  per  unit 
volume   by  the  foreign   sources  of   energy.       For,  multiplying 


DY.NAMIC    EQUATIONS   OF   ELECTROMAGNETIC    FIELDS.         77 

these  equations  by  the  energetic  field  intensities  and  adding,  we  get 
(^a)  a^  •  c  +  b  •  k  =  a^  •  curl  b  -  b^  •  curl  a. 

The  left  hand  member  gives  the  rate  at  which  this  energy  is  sup- 
plied. The  discussion  of  the  right  hand  member  therefore  will 
show  how  the  energy  supplied  is  stored,  transformed,  or  moved  to 
other  places.  In  this  discussion  we  shall  follow  the  method  mdi- 
cated  by  Heaviside.* 

4.  Storage,  Transformation,  and  Motion  of  the  Energy.  —  io 
examine  the  right  hand  member  of  the  equation  we  express  the 
energetic  field  intensities  as  the  differences  of  the  actual  and  the 
induced  field  intensities, 

a^  =  a„  -  a,         b^  =  b,^  -  b. 
The  equation  of  activity  then  takes  the  form 
^a)     a  .  c  +  b^  •  k  =  a ,  curl  b  -  b„  •  curl  a  -  a  ■  curl  b  +  b  •  curl  a. 
For  the  last  two  forms  we  write,  according  to  a  well  known  vec- 
tor formula, 
(■J)  _  a  •  curl  b  +  b  •  curl  a  =  div  (a  x  b). 

In  the  first  term  on  the  right  hand  side  of  equation  {a)  we  in- 
troduce for  curl  b  the  developed  expression  for  the  electric  current, 

(III.,  10,  h.).     Thus 

^.  1 

(c)    a„curlb=a,,/^  +  a„  curl(AxV)+a„VdivA+ya,A. 

Remembering  that  A  =  ota,,  we  find  easily, 

K-Tt  =  ^^'^^r  =  ''^-dt-^^^'dt 

8  da         .,  da 

or  finally 

*0  Heavlide  :  On  the  forces,  stresses  and  fluxes  of  energy  in  the  electro- 
n^agneti!  field.  Transactions  of  the  Koyal  Society,  London,  1892.  Electrical 
papers,  Vol.  11,  p.  52L 


78  FIELDS    OF    FORCE. 

cA  ^    /  2  ^^ 

Now  we  have  in  general  (III.,  10,  h^) 
da       da 

And  if  we  suppose  that  the  moving  individual  element  does  not 
change  the  value  of  its  inductivity  as  a  consequence  of  the  mo- 
tion, we  have  dajdt  =  0,  and 

da 

And  therefore 

dA       d 
(c^  a  ■  "^-  =  ^,  (iA  ■  a  )  -  V  •  ia- Va. 

Passing  to  the  next  term  in  (c),  we  can  transform  it  by  the  vec- 
tor formula  (i),  writing  a^  for  a  and  A  x  V  for  b.     Thus 

a^  •  curl  (Ax  V)  =  A  x  V  ■  curl  a^  —  div  [a_  x  (A  x  V)]  . 

In  the  first  right  hand  term  we  interchange  cross  and  dot,  and 
change  the  order  of  factors  by  cyclic  permutation.  In  the  second 
term  we  develope  the  triple  vector  product  according  to  the  well- 
Uiiowii  foi-miilii;  we  have  then 

(cj  a,  •curl(Ax  V)  =  V  (cuvlajx  Afdiv  [(a,   A)V-(a^  •  V)A]  . 

Substituting  (r-.)  and  (c^)  in  (c)  we  get 

d  1 

{d\     ^" '  "''""^  ^  ^  c}t  ^^^  ■  ^"^  +  T^A  •  a«  +  V  •  [  (div  A)a„ 

-  Kvo!  +  (curl  aj  X  A]  -f  div  {(A •  ajV  -  (a,  •  V)A] . 

In  exactly  the  same  way,  introducing  the  full  expression  for  the 
magnetic  current,  we  have 

(,)  -b,curla  =  ^(P-bJ+   ^,Bb,  +  V{(divB)b„ 

-  ib;^V/3  +  (curl  bj  X  B}  +  div  [(B  •  bjV  -  (b„  •  V)B}. 


DYNAMIC    EQUATIONS    OF    ELECTROMAGNpync    FIELDS.       79 

Tlie   developments  (h),  {d),  and  (r)  are  now   introduced   in  (^n). 
Suitably  distributing  the  terms,  we  get 

d 
ac+bk=^{iAa  +iBb} 

1  1 

+  ^A  a,  +  ^,B  b„ 

^'^  )  +V  •  {(div  A)a^  —  ia;v^  +  (curl  aj  x  A} 

+  V  •  { (div  B)b^  -  ib;  v/3  +  (curl  bj  x  B] 
+  div  [ax  b  +  |(A ■  a,  +  B •  bjV} 
+  div  {_(a,^-  V)A  +  J(A  -  aJV  -  (b^-  V)B  +  K^'^J^  }, 

which  is  the  completely  developed  form  of  the  equation  of  activity. 

The   first  member  gives,  as  we  have  said,  the  rate  of  supply  of 

electromagnetic  energy  per  unit  volume,  and  the  second  member 

shows  how  the  energy  supplied  is  used.     Taking  one  term  after 

the  other  in  each  line,  the  common  interpretation  of  them  is  this  : 

The  first  term 

d 

~dt 


{iA-a^+  p-b, 


gives  the  part  of  the  energy  supplied  which  is  simply  stored  as 
electric  and  magnetic  energy  in  the  unit  volume.    The  second  term 

1  1 

^A  a„+  ^,B  b„ 

gives  the  part  of  the  energy  supplied  which  is  wasted  as  heat, 
according  to  Joule's  law,  the  waste  due  to  the  fictitious  magnetic 
conduction  current  being  also  formally  included. 

The  following  two  terms  contain  the  velocity  V  of  the  moving 
material  element  of  volume  as  a  scalar  factor.  As  the  equation  is 
an  equation  of  activity,  the  other  factor  must  necessarily  be  a 
force,  in  the  common  dynamic  sense  of  this  word,  referred  to 
unit  volume  of  the  moving  particle.  These  factors  are  then  the 
forces  exerted  by  the  electromagnetic  system  against  the  exterior 


80  FIELDS    OF    FORCE. 

forces,  the  factor  of  the  first  terra  being  the  mechanical  force  de- 
pending on  the  electric  field,  and  the  factor  of  the  second  term 
being  the  force  depending  upon  the  magnetic  field, 

f  =  (div  A)a^  —  i^f  Va  +  Ccurl  a  J  x  A, 
^^^  f,„  =  (div  B)b„  -  i-bf  V/3  +  (curl  bj  x  B. 

The  first  of  the  two  terms  of  (/)  which  have  the  form  of  a  di- 
vergence gives,  according  to  the  common  interpretation,  that  part 
of  the  energy  supplied  which  moves  away.  There  are  two  reasons 
for  this  uiotion  of  energy,  first,  the  radiation  of  energy,  given  by 

the  Poynting-flux 

a  X  b, 

and  second,  the  pure  convection  of  electromagnetic  energy,  given 

by  the  vector 

^(A-a„  +  B-bjV, 

which  is  simply  the  product  of  the  energy  per  unit  volume  into 
the  velocity. 

Finally,  the  last  term  gives,  according  to  the  common  interpre- 
tation, that  part  of  the  energy  supplied  which,  in  terms  of  the 
theory  of  the  motion  of  energy,  moves  away  in  consequence  of  the 
stress  in  the  medium  which  is  the  seat  of  the  field,  the  flux  of 
energy  depending  upon  this  stress  being  given  by  the  vector 

-  iK  •  V)A  +  1  ( A  •  aJV  -  (b„  •  V)B  +  i(B  •  bJV, 

whose  divergence  appears  in  the  equation  of  activity.  For  this 
flux  of  energy  may  be  considered  as  that  due  to  a  stress,  the  com- 
poticnt  of  which  against  a  plane  whose  orientation  is  given  by 
the  unit  normal  N  is 

a„(A  ■  N)  -  (1 A  ■  aJN  +  b^B  •  N)  -  (iB  ■  bJN. 

This  stress  splits  up  into  an  electric  and  a  magnetic  stress.  And, 
in  the  case  of  isotropy,  which  we  assume,  the  first  of  these 
stresses  consists  of  a  tension  parallel  to,  and  a  pressure  perpen- 


DYNAMIC    EQUATIONS    OF    ELECTRO^IAGNETIC    FIELDS.        81 

dicular  to  the  lines  of  electric  force,  in  amount  equal  to  the  elec- 
tric energy  per  unit  volume  ;  the  second  consists  of  a  tension  and 
pressure  bearing  the  same  relation  to  the  magnetic  lines  of  force 
and  magnetic  energy  per  unit  volume.  This  is  seen  when  the  unit 
normal  N  is  drawn  first  parallel  to,  and  then  normal  to  the  corre- 
sponding lines  of  force. 

The  theory  thus  developed  may  be  given  with  somewhat  greater 
generality  and  with  greater  care  in  the  details.  Thus  the  aniso- 
tropy  of  the  medium,  already  existing,  or  produced  as  a  conse- 
quence of  the  motion,  can  be  fully  taken  into  account,  as  well  as 
the  changes  produced  by  the  motion  in  the  values  of  the  induc- 
tivities  and  in  the  values  of  the  energetic  vectors.  On  the  other 
hand,  there  exist  differences  of  opinion  with  regard  to  the  detads 
of  the  theory.  But  setting  these  aside  and  considering  the  ques- 
tion from  the  point  of  view  of  principles,  is  the  theory  safely 
founded  ?  If  we  knew  the  real  physical  significance  of  the  electric 
and  magnetic  vectors,  should  we  then  in  the  developements  above 
meet  no  contradictions  ? 

This  question  may  be  difficult  to  answer.  The  theory  must 
necessarily  contain  a  core  of  truth.  The  results  which  we  can 
derive  from  it,  and  which  depend  solely  upon  the  pnnciple  ot  the 
conservation  of  energy  and  upon  the  expression  of  the  electro- 
magnetic energy,  so  far  as  this  expression  is  empirically  tested, 
must  of  course  be  true.  But  for  the  rest  of  the  theory  we  can 
only  say,  that  it  is  the  best  theory  of  the  dynamic  properties  ot 
the  electromagnetic  field  that  we  possess. 

5  The  Forces  in  the  Electromagnetic  Fiehl-^yhai  particularly 
interests  us  is  the  expression  for  the  mechanical  forces  in  the  held, 
(4  a)  As  the  expressions  for  the  electric  and  the  magnetic  force 
have  exactly  the  same  form,  it  will  be  sufficient  to  consider  one 
of  them.     Let  us  take  the  magnetic  force, 

f  =  (div  B)b„  -  IKV^  +  (curl  b.)  x  B. 
This  is  a  force  per  unit  volume,  and  if  our  theory  is  correct,  this 
expression  should  give  the  true  distribution  of  the  force  acting  upon 

11 


82  FIELDS    OF    FORCE. 

the  elements  of  volume,  and  not  merely  the  true  value  of  the  re- 
sultant force  upon  the  whole  body.  The  significance  of  each  term 
is  obvious.  The  first  term  gives  the  force  upon  the  true  magnet- 
ism, if  this  exists.  It  has  the  direction  of  the  actual  field  intensity, 
and  is  equal  to  this  vector  multiplied  by  the  magnetism.  The 
second  term  depends  upon  the  heterogeneity  of  the  bodies,  and 
gives,  therefore,  the  force  depending  upon  the  induced  magnetism. 
The  elementary  force  which  underlies  the  resultant  forces  observed 
in  the  experiments  of  induced  magnetism  should  therefore  be  a 
force  which  has  the  direction  of  the  gradient,  —  v/3,  of  the  induc- 
tivity  /3,  and  which  is  equal  in  amount  to  the  product  of  this  gra- 
dient into  the  magnetic  energy  per  unit  volume.  When  we  consider 
a  body  as  a  whole,  the  gradient  of  energy  will  exist  principally  in  the 
layer  between  the  body  and  the  surrounding  medium.  It  will  point 
outwards  if  the  body  has  greater  inductivity  than  the  medium, 
but  its  average  value  for  the  whole  body  will  be  nil  in  every  direc- 
tion. But  the  force,  which  is  the  product  of  this  vector  into  half 
the  square  of  the  field  intensity,  will  therefore  have  greater  aver- 
age values  at  the  places  of  great  absolute  field  intensity,  quite 
irrespective  of  its  direction.  Hence,  the  body  will  move  in  the 
direction  which  the  inductivity  gradient  has  at  the  places  of  the 
greatest  absolute  strength  of  the  field,  i.  e.,  the  body  will  move 
in  the  direction  of  increasing  absolute  strength  of  the  field.  And, 
in  the  same  way,  it  is  seen  that  a  body  which  has  smaller  induc- 
tivity than  the  surrounding  field  will  move  in  the  direction  of 
decreasing  absolute  strength  of  the  field.  The  expression  thus 
contains  Faraday's  well  known  qualitative  law  for  the  motion  of 
magnetic  or  diamagnetic  bodies  in  the  magnetic  field. 

The  third  term  of  the  equation  contains  two  distinct  forces, 
which,  having  the  same  form,  are  coml)ined  into  one.  Splitting 
the  actual  field  intensity  into  its  induced  and  energetic  parts  and 
treating  the  curl  of  the  vector  in  the  same  way,  we  get 

curl  b   =  curl  b  -}-  curl  b  =  c  4-  c  , 

where  c  is  f/ic  (rue  electric  current,  and  c^  the  fictitious  current,  by 


DYNAMIC    EQUATIONS    OF    EI.ECTRO MAGNETIC    FIELDS.        83 

which,  according  to  Ampere's  theory,  the  permanent  magnetism 
may  be  represented.  The  last  term  of  the  expression  for  the  force 
therefore  splits  into  two, 

(curl  hj  X  B  =  c  X  B  +  c^  x  B, 

where  the  first  term  is  the  well  known  expression  for  the  force  per 
unit  volume  in  a  body  carrying  an  electric  current  of  density  c. 
The  second  terra  gives  the  force  upon  permanent  magnetization, 
and  according  to  the  theory  developed,  this  force  should  be  the 
same  as  the  force  upon  the  equivalent  distribution  of  electric 
current. 

6.  The  Resultant  Force.  —  As  we  have  remarked,  our  develop- 
ments may  possibly  contain  errors  which  we  cannf)t  detect  in  the 
present  state  of  our  knowledge.  The  value  found  for  the  elemen- 
tary forces  may  be  wrong.  But  however  this  may  be,  we  know  th'n^ 
with  perfect  certainty  ;  if  we  integrate  the  elementary  forces  for 
the  whole  volume  of  a  body,  we  shall  arrive  at  the  true  value  of 
the  resultant  force  to  which  the  body  as  a  whole  is  subject.  For 
calculating  this  resultant  force,  we  come  back  to  the  results  of 
the  observations  which  form  the  empirical  foundation  of  our 
knowledge  of  the  dynamic  properties  of  the  electromagnetic  field. 
A  perfectly  safe  result  of  our  theory  will  therefore  consist  in  the 
fact  that  the  expression 

(rt)      F  =  /(div  B)h  Jt  -  /^b^V/Sc/r  +  /(curl  bj  x  Bch, 

where  the  integration  is  extended  over  a  whole  body,  gives  the  true 
value  of  the  resultant  force  upon  the  body.  By  a  whole  body, 
we  understand  any  body  surrounded  by  a  perfectly  homogeneous 
gaseous  or  fluid  dielectric  of  the  constant  inductivity  ^„,  which  is 
itself  not  the  seat  of  any  magnetism  M,  of  any  energetic  mag- 
netic flux  B^,  or  of  any  electric  current  c.  To  avoid  mathe- 
matical prolixity  we  suppose  that  the  properties  of  the  body 
change  continuously  into  those  of  the  ether,  the  layer  in  which 
these  changes  take  place  being  always  considered  as  belonging  to 
the  body.     Thus  at  its  surface  the  body  has  all  the  properties  of 


84  FIELDS    OF    FORCE. 

the  ether.  By  this  supposition,  we  shall  avoid  the  introduction 
of  surface  integrals,  which  usually  appear  when  transformations  of 
volume  integrals  are  made. 

By  transformations  of  the  integrals  we  can  pass  from  the  above 
expression  for  the  resultant  force  to  a  series  of  equivalent  ex- 
pressions. To  find  one  of  these  new  expressions  we  split  the 
actual  field  intensity  into  its  two  parts, 

b   =  b  +  b  , 
and  we  get 

(6)      F  =  /(div  B)b(ZT  -  /  Jb- VySr/r  +  /(curl  b)  x  Bdr  +  J, 
where 

J  =  r(div  B)b  (h  -  f(h  ■  b )  V/SJt 

-  f^hlv^ch  +  /(curlb)  X  Bdr. 

To  reduce  the  expression  for  J  we  consider  the  first  term. 
Transforming  according  to  well  known  formulse,  we  get 

/(div  B)b/?T  =  -  /Bvb//T  =  -  jBh^vdr  -  /(curl  b  )  x-  Bdr. 

Substituting,  we  get  J  reduced  to  three  terms, 

{h")  ]=-  fB\vdT  -  /(b  •  bj  v/3r7T  -  /lb;  v/3f7T. 

Introducing  in  the  first  of  these  integrals  B  =  /Sb  +  /3b ,  we  get 

-  /Bb^vr/r  =  -  //3(bb  v)fZT  -  f^(hh^v)dT, 

in  which  we  have  to  remember  that  the  operator  v  works  only 
upon  the  vector  immediately  preceding  it.  In  the  first  of  the 
two  integrals  of  the  right  hand  member  we  join  the  scalar  factor  /3 
with  the  vector  b ,  upon  which  v  works,  remembering  /8b^  =  B^. 
A  term  containing  VyS  must  then  be  subtracted.  The  second 
integral  we  can  change,  letting  the  operator  v  work  upon  both 
factors.     Then 

-  /Bb.V'/T  =  -  /bB^vr/r  +  /(bb,)v/3r7T  -  /|/3vbV7T. 


DYNAMIC   EQUATIONS   OF    ELECTROMAGNETIC   FIELDS.        8') 

Finally,  integrating  the  last  term  by  parts  and  remembering  that 
b^  disappears  at  the  surface  of  the  body, 

-  fBh^vdr  =  -  /bB,  VfZr  +  J"(b   bj V^f/r  +  /^b; V  A?t. 
Substituting  this  in  (6"),  we  get  simply 
(6'")  J  =  -fhBydr. 

This  leads  to  the  expression 
(c)    F  =/(div  B)bdr  -fWv/Sdr  +f[cm'\  b)  x  Bdr  -JhB^  V  dr 

for  the  resultant  force.  The  four  terras  give  the  forces  depending 
upon  the  true  magnetism,  the  induced  magnetism,  the  electric  cur- 
rent distribution,  and  the  i^erraanent  magnetization  respectively. 
The  resultant  force  is  represented  here  by  a  system  of  elementary 
forces,  given  by 

1,  =  (div  B)b  -  WvB  -bB^V  +  (curl  b)  x  B. 
These  elementary  forces  must  be  considered  as  fictitious  if  the 
expression  found  above  represents  the  true  values  of  the  ele- 
mentary forces.  But  if  our  developments  have  not  been  altogether 
trustworthv,  the  reverse  might  also  be  the  case,  or  else  none  of 
them  may  give  the  true  values  of  the  elementary  forces,  while  both 
of  them  give  the  true  values  of  the  resultant  forces. 

7.  Other  Forms  for  the  Resultant  Force.  — In  writing  the  ex- 
pression for  the  resultant  force  we  have  hitherto  used  scalars  and 
vectors  of  a  fundamental  nature.  By  the  introduction  of  certani 
auxiliary  scalars  or  vectors  the  expression  for  the  resultant  force 
may  be  brought  to  forms  of  remarkable  simplicity.  But  as  this 
is  obtained  at  the  cost  of  the  introduction  of  artificial  quantities, 
the  possibility  that  the  expressions  under  the  integral  signs  repre- 
sent the  real  elementary  forces  is  lost. 

The  transformation  to  these  simple  forms  of  the  expression  for 
the  resultant  force  depends  upon  the  introduction  of  a  vector  B , 
defined  by  the  equation 
(a)  B  =  /3„b  +  B, 


86  FIELDS    OF    FORCE. 

This  has  the  form  of  the  true  equation  of  connection,  excej3t  that 
the  constant  inductivity  /3^^  of  the  ether  is  introduced  instead  of 
the  true  inductivity  of  the  body.  B^  is  therefore  a  virtual  ener- 
getic flux,  to  compensate  for  our  leaving  out  of  consideration  the 
variations  of  the  inductivity.  This  is  the  well  known  artifice  of 
Poisson's  theory  of  induced  magnetism,  which  enables  us  to  treat 
the  induced  magnetism  as  if  it  were  permanent. 

To  introduce  this  vector  into  the  expression  for  the  resultant 
force  we  first  remark  that  in  the  second  integral  of  the  expression 
(6,  c)  we  can  write  ^  —  ^^  instead  of  yS.  Performing  the  integra- 
tion by  parts  throughout  the  whole  volume  of  the  body  and  remem- 
bering tliat  /3  —  /Sjj  disappears  at  the  surface  of  the  body,  we  get 

-  /AbV/Sf?T  =  _/ibV(/3  -  y8,>?T 

=     /(^-^JbbvrfT. 

In  like  manner,  the  transformation  by  parts  of  the  integral  in  the 
expression  (0,  e)  expressing  the  force  upon  permanent  magnetism 
gives 

—  JbB^  V  (It  =  jBhsydr. 

The  integrals  for  the  temporary  and  the  permanent  force  may  now 
be  added,  and  remarking  that  equation  («),  in  connection  with 
tlie  fundamental  equation  of  connection,  gives  B  =  (/3  —  /S,,)b  -|-  B^, 
we  get 

-  JWv^dr-JhBydT  =  jB^bVf?T. 

The  substitution  of  tliis  in  ((J,  c)  gives  the  following  more  com- 
pact form  of  tlie  expression  for  the  resultant  force 

(h)  F  =  /(div  B)b^/T  -h  /b  ,bV(/T  +  J(curl  b)  x  B(?t. 

Here,  the  resultant  force  seen)s  to  come  from  an  elementary  force 
fg  =  (div  Bp  +  B^bv  -f  (curl  b)  x  B. 


DYNAMIC    EQL^ATIONS   OF    ELECTROMAGNETIC   FIELDS.        87 

A  still  shorter  form  of  the  resultant  force  and  of  the  corre- 
sponding fictitious  elementary  force  may  be  found  as  follows. 
According  to  a  well  known  vector  formula,  we  can  write 

jB^hvdr  =  Jb,  vb(7T  -  /(curl  b)  x  B/h. 

Transforming  the  first  integral  of  the  second  member  according 
to  a  well  known  formula  and  remembering  that  B^.  =  0  at  the 
surface  of  the  body,  we  get 

fB.hvdr  =  —  /(div  B,)bf/T  —  /{curl  b)  x  B.dr. 

Introducing  this  expression  and  remarking  that,  according  to  (c), 
div  B  =  /3^  div  b  +  div  B^.,  we  get 

F  =  ^^  /(div  b)bf/T  +  /3^,  /(curl  b)  x  bdr, 

which  is  the  most  concise  form  of  the  expression  for  the  resultant 
force.     It  is  expressed  here  by  a  fictitious  elementary  force 

f^  =  yS„  (div  b)b  +  yS,  (curl  b)x  b. 

The  divergence  of  the  field  intensity,  which  appears  here,  is 
called  the  free  density  of  magnetism.  The  force  upon  true  mag- 
netism, upon  permanent  magnetic  polarization,  and  upon  induced 
magnetism  can  be  condensed  into  one  expression,  and  the  whole 
force  is  expressed  in  an  exceedingly  simple  way  by  the  field  in- 
tensity, its  divergence,  its  curl,  and  the  inductivity  of  the  sur- 
rounding medium. 

8.  Eesimie  —  It  will  be  convenient  on  account  of  the  following 
lectures  to  sum  up  the  fundamental  equations  for  the  stationary 
electric,  and  the  stationary  magnetic  field.  Using  for  the  descrip- 
tion of  the  fields  the  vectors  of  scheme  III.,  and  in  some  cases 
even  the  artificial  vectors  A^.  or  B^.  (lY.,  7,  a),  we  have  first  a  set 
of  equations  of  connection,  by  use  of  which  we  introduce  in  the 
fundamental  equations  the  vector  wanted  for  any  special  purpose. 
Of  these  equations  of  connection  we  note  the  following,  referring 
for  more  special  cases  to  the  complete  system  (III.,  7,  a). 


88  FIELDS   OF    FORCE. 

Electric  Magnftic 

A  =  aa„,  B  =  ^b„, 

(A)  =  aa  +  A ,  =  ^b  +  B^, 

=  a„a  +  A,,  =  ^,b  +  B,. 

Then  we  have  the  proper  equations  of  the  fields,  which  express 
the  relation  between  the  field  intensity  and  the  current  density, 

(^B)  curl  a  =  —  k,  curl  b  =  c. 

Finally,  we  have  the  equations  of  definition  for  the  density  of 
electrification,  or  of  magnetism, 

( C)  div  A  =  B,         div  B  =  M. 

To  complete  the  geometric  description  of  the  field  w^e  have  finally 
a  number  of  special  conditions  which  are  fulfilled  in  the  free  ether, 
namely, 

(A)  «  =  «o'  /3  =  ^,„ 

{JD,)  -^=0,  31=0, 

(A)  k=0,           c  =  0,        • 

(A)  A^=0.  B^  =  0- 

This  set  of  equations  gives,  in  the  sense  of  the  theorems  (HI.,  16), 
a  complete  description  of  the  geometry  of  the  fields. 

Our  knowledge  of  the  dynamics  of  the  field  is  less  complete. 
According  to  the  analysis  of  Heaviside,  we  have  reason  to  believe 
that  the  elementary  force  in  the  field  per  unit  volume  is  given  by 
the  expression 

f      =  (div  A)a  —  ia^  V^  +  (curl  a  )  x  A, 
f„, ,  =  (div  B)b.  -  lb; v/3  +  (curl  bj  X  B. 
But  other  forms  are  not  excluded,  and  we  may  have 

f,  .,  =  (div  A)a  —  ?,a- v^:  +  (curl  a)  x  A  —  aA  v, 
(E.)  .-      V  y         -  V  J 

f,„^  ^  =  (div  B)b  -  1  b-VyS  +  (eurl  b)  x  B  -  bB^v. 


DYNAMIC    EQUATIONS    OF    ELECTROM AGNPync    FIELDS,        S!l 

Our  reliable  knowledge  is  reduced  to  this  —  we  get  on  integrat- 
ing any  of  these  forces  for  a  whole  body  the  resultant  force  which 
produces  the  motion  of  the  whole  body.  The  same  value  of  the 
resultant  force  may  also  be  found  from  other  purely  artificial  dis- 
tributions of  the  elementary  force,  for  example, 


or 


f^  ^  =  (div  A)a  -f-  A^  av  -I-  (curl  a)  x  A, 
4  3  =  (div  B)b  +  B,.b V  +  (curl  b)  x  B, 

t,  4  =  ^o(f^^v  ^)^  +  a^(curl  a)  X  a, 
f „,  ^  =  /3^(div  b)b  +  /3,(curl  b)  X  b. 


12 


y. 

GEOMETRIC  AND  DYNAMIC   PliOPERTIES  OF  THE 
HYDEODYNAMIC  FIELD.      GENERAL  DEM- 
ONSTRATION OF  THE  ANALOGY  TO 
THE    STATIONARY    ELECTRO- 
MAGNETIC FIELDS.    • 

1 .  Prelhiuii(u-y  Remarks.  —  Our  preliminary  investigations, 
based  on  elementary  reasoning  and  experiment,  have  already 
given  the  general  feature  of  the  analogy,  which  we  are  now  going 
to  examine  more  closely.  According  to  these  preliminary  results, 
we  have  no  reason  to  look  for  an  analogy  extending  beyond  the 
phenomena  termed  stationary.  The  main  feature  of  the  analogy 
is  given  by  the  correspondence  : 

flux velocity, 

field  intensity specific  momentum, 

inductivity mobility  (specific  volume). 

To  facilitate  the  comparison  of  the  fields  I  shall  denote  the 
hydrodynamic  quantities  by  the  same  letters  as  the  corresponding 
electrical  quantities.  The  symmetry  in  the  properties  of  the  elec- 
tric and  magnetic  fields  will  make  it  possible  to  pass  at  once  from 
the  comparison  with  the  electric  field  to  the  comparison  with  the 
magnetic  field. 

2.  The  I Iijdr adynamic  Equations.  —  The  basis  of  our  investi- 
gation will  be  the  hydrodynamic  equations,  of  which  there  are 
two;  the  scalar  ecpiation  for  the  conservation  of  the  mass,  generally 
called  the  equation  of  continuity,  and  the  vector  equation  of  motion. 

a  being  the  specific  volume  of  the  fluid,  A  the  vector  velocity, 
and  fJ'df  representing  the  individual  time-differentiation,  the  equa- 
tion of  continuity  may  be  written 

(a)  =  div  A. 

^  ^  a  at 

90 


PROPERTIES    OF    THE    HYDRODYNAMIC    FIELD.  91 

The  first  member  is  the  velocity  of  expansion  per  unit  volume 
of  the  moving  fluid  particle,  expressed  through  the  effect  of  this 
expansion  upon  the  specific  volume,  or  the  volume  of  unit  mass. 
The  second  member  is  the  same  velocity  of  expansion  expressed 
through  the  distribution  of  velocity  in  the  fluid.  The  equality 
of  these  two  expressions  of  the  same  velocity  of  ex])ansion  insures 
the  conservation  of  the  mass  during  the  motion  of  the  fluid. 

Now  f  being  the  exterior  force  acting  per  unit  volume  of  the 
moving  fluid  masses,  and  p  the  pressure  in  the  fluid,  the  vector 
equation  of  motion  may  be  written 

(P)  adt='-^^- 

The  first  member  is  the  product  of  the  density,  l/ot,  of  the  moving 
particle  into  its  acceleration,  dkjdt,  and  the  second  member  gives  the 
vector  sum  of  the  forces  per  unit  volume  acting  upon  it.  These 
forces  are  the  exterior  force  f,  and  the  force  due  to  the  pressure, 
—  Vp,  generally  called  the  gradient. 

In  the  use  of  these  equations  it  is  always  to  be  remembered  that 
the  individual  differentiating  symbol  djdt  is  related  to  the  local 
differentiating  symbol  djdt  by  the  Eulerian  expansion 

d        d 

W  J^  =  a^"  +  ^^- 

These  equations  do  not  give  the  geometry  and  the  dynamics  of 
the  hydrodynamic  field  as  separate  theories.  They  contain  the 
properties  of  the  fields  viewed  from  one  central  point,  from  which 
their  geometric  and  dynamic  properties  seem  perfectly  united.  It 
will  be  our  problem  to  artificially  separate  from  one  another  cer- 
tain geometric  and  certain  dynamic  properties,  m  order  to  be 
able  to  carry  out  the  comparison  with  those  other  fields  which  we 
know  onlv  as  the  result  of  an  inspection  from  without,  an  inspec- 
tion which  has  allowed  us  only  to  recogni/e  two  separate  sides 
of  their  properties,  without  any  deeper  insight  into  their  true 
relations. 


92  FIELDS    OF    FORCE. 

3.  Equation  of  Continuity  —  Equation  for  the  Density  of  Electri- 
fication. —  The  equation  of  continuity  has  the  form  of  one  of  the 
fundamental  equations  of  the  electric  field.  To  show  this  we  have 
only  to  represent  the  v'elocity  of  expansion  per  unit  volume, 
\ja  dajclt,  by  a  single  letter  E,  and  obtain  the  equation  corre- 
sponding to  (IV.,  8,  C), 

div  A=  E, 

which,  in  the  interpretation  of  the  symbols  for  the  electrical  case,  is 
the  equation  which  gives  the  density  of  electrification  in  the  elec- 
tric field. 

4.  Transformation  of  the  Dynamic  Equation.  —  The  dynamic 
equation  does  not  in  its  original  form  show  any  resemblance  to 
any  of  the  equations  of  the  electric  field.  Some  simple  transforma- 
tions will,  however,  bring  out  terms  of  the  same  form  as  appear  in 
the  dynamic  equations  of  the  electric  field. 

To  show  this  let  us  first  introduce  instead  of  the  velocity  A  the 
actual  specific  momentum,  a  ,  according  to  the  equation 

(a)  '  A  =  aa  . 

The  equation  of  motion  then  takes  the  form 

da        1  da 

-di  +  a  dt  *"  =  *  -  '^1'' 
or,  according  to  the  equation  of  continuity  (2,  a), 

^«-f  (divA)a„  =  f-v;^. 

In  the  left  hand  member  we  have  the  term  (div  A)a^^,  the  analogue 
of  which  appears  in  the  expression  f^  for  the  elementary  forces  in 
the  electric  field  (IV.,  8,  A\).  It  is  the  elementary  force  acting 
upon  the  true  electrification,  div  A. 

Further  simple  transformations  bring  in  the  other  corresponding 
terms  appearing  in  the  expression  for  f,  for  the  elementary  forces 
in  the  electric  field.     Using  the  Eulerian  expansion,  we  first  get 

da 

^'^  +  Ava,,  +  (div  A)a^  =  f  —  v;^, 


PROPERTIES   OF   THE    HYDRODYNAMIC    FIELD. 


93 


and  then  transforming  the  second  left  hand  term  according  to  a 
well  known  vector  formula,  we  have 

^+  Aa  V  +  (curl  a  )  x  A  +  (div  A)a,  =  f  -  Vy>. 

Now,  the  term  (curl  aj  x  A  has  appeared,  which  correspondingly 
appears  in  the  expression  (lY,  8,  E,)  for  the  force  in  the  electric 
field,  representing  in  one  term  the  force  exerted  upon  permanent 
polarization  and  upon  magnetic  current. 

According  to  (a),  the  second  term  in  the  left  hand  member  may 

be  written  ^ 

Aa,v  =  ^a,a,v  =  J'^va;, 

or  finally, 

Aa„v  =  V(j^a;)  -  iK^a. 

Substituting  this  above,  we  have 

^^«  +  V  (i«a;;)  -  |a;Vot  +  (curl  aj  x  A  +  (c^v  A)a,  =  f  -  VyJ, 

giving  us  all  the  corresponding  terms  contained  in  the  expression 
for  the  force  (IV.,  8,  E;)  in  the  electric  field. 

5.  Separation  of  the  Equation  of  Motion.--\Ne  thus  seem  to  have 
found  some  relation  between  the  hydrodynamic  equation  and  the 
equation  giving  the  dynamics  of  the  electric  field.  But  we  still 
have  the  geometry  and  the  dynamics  of  the  hydrodynamic  field 
united  in  one  set  of  equations.  To  make  the  first  step  to^^^rds 
the  separation  of  certain  geometric  and  dynamic  properties  from 
one  another  we  have  to  consider  the  hydrodynamic  field  as  the  sum 
of  two  partial  fields,  just  as  we  consider  the  electric  field  as  the  sum 
of  two  partial  fields,  the  induced  and  the  energetic  field.  Let  us 
represent  the  vector  a„,  the  actual  specific  momentum,  as  the  sum 
of  two  vectors  a  and  a^,  thus 

(a)  a,  =  a  +  a. 

The  equation  then  develops  into 

-'     ''       ^*  +(divA)a.  =  f-vp. 


94  FIELDS   OF    FORCE. 

Now  we  have  the  right  to  submit  one  of  the  auxiliary  vectors, 
say  a,  to  a  condition.  Let  this  condition  be  that  it  shall  satisfy 
the  equation 

The  other  vector  will  then  have  to  satisfy  the  equation 

aa 

(c)  — "  =  f  —  (div  A)a^^  +  |a; v«  —  (curl  a,)  x  A. 

6.  Geo'indric  Properfy  of  the  Induced  Motion.  —  We  have  thus 
introduced  the  consideration  of  two  fields,  which  superimposed 
upon  each  other  represent  the  actual  hydrodynamic  field.  But  the 
equations  of  both  partial  fields  are  still  dynamic  equations.  How- 
ever, from  one  of  them  we  can  at  once  proceed  to  a  purely 
geometric  equation.  For  taking  the  curl  of  e([uation  (b)  and 
changing  the  order  of  the  operations  d/dt  and  curl,  we  get 

d      , 

^^curl  a=  0. 
'  ct 

To  complete  the  nomenclature  I  will  call  the  curl  of  the  velocity 
the  kinevKific,  and  the  curl  of  the  specific  momentum  the  dynamic 
vortex  demiti/.  The  dynamic  vortex  density  is  thus  invariable  at 
every  point  of  space.  Integrating  with  respect  to  the  time  and 
writing  —  k  for  the  constant  of  integration,  we  get 

(d)  curl  a  =  —  k, 

which  expresses  the  local  conservation  of  the  dynamic  vortex  den- 
sity. As  regards  its  form,  this  is  the  same  equation  which  in 
the  electric  interpretation  of  the  symbols  expresses  the  relation  be- 
tween the  electric  field  intensity  a  and  the  magnetic  current  k  (IV., 
8,  B).  And,  as  the  conservation  of  k  is  local,  equation  (d)  cor- 
responds exactly  to  the  equation  for  the  electric  field  for  the  cases 
of  magnetic  currents  which  are  stationary  both  in  space  and  in  time. 

7.  Fnndamental  (k'ometric  Properties  of  the  Jli/drodi/namic  Field. 
— We  have  thus  succeeded  in  representing  the  hydrodynamic  field 


PROPERTIES    OF    THE    HYDRODYNAMIC    FIELD.  95 

as  the  sum  of  two  partial  fields.  Writing  A^  =  ota^,  we  have  for  the 
vectors  introduced  the  equation  of  connection 

(J.)  A  =  aa^  r=  aa  -f  A^. 

Then  the  induced  field  described  by  a  has  the  property  of  local 
conservation  of  the  dynamic  vortex, 

(i?)  curl  a  =  —  k, 

while  from  the  field  of  the  actual  velocity  we  calculate  the  veloc- 
ity of  expansion  per  unit  volume,  E,  from  the  equation 

(C)  divA  =  ^. 

In  form,  these  equations  are  precisely  the  fundamental  eijuations 
for  the  geometric  properties  of  tlie  stationary  electric  field. 

8.  Bodies  and  Fundamental  Fluid.  —  To  complete  the  investiga- 
tion of  the  geometric  properties  we  shall  have  to  examine  whether 
we  can  introduce  conditions  corresponding  to  the  supplementary 
conditions  (IV.,  8,  D).  The  introduction  of  conditions  of  this 
nature  for  the  fluid  system  evidently  involves  the  distinction  be- 
tween certain  limited  parts  of  the  fluid,  which  we  have  to  com- 
pare with  material  bodies,  and  an  exterior  unlimited  part  of  the 
fluid,  which  we  have  to  compare  with  the  free  ether.  The  part  of 
the  fluid  surrounding  the  fluid  6ot?«w  we  shall  call  the  fundameufal 
fluid. 

Introducing  the  condition 

(A)  ^  =  "'«' 

where  a^  is  constant,  we  simply  require  the  fundamental  fluid 

to  be  homogeneous.     Introducing  the  condition 

(A)  ^=«' 

we  require  it  to  be  incompressible.     There  is  nothing  which  pre- 
vents us  from  introducing  the  additional  condition 

(A)  ^  =  ^' 

for,  at  every  point  of  space  the  dynamic  vortex  has,  according  to 


96  FIELDS    OF    FORCE. 

the  fundamental  equation  (B),  a  constant  value.  We  are  there- 
fore free  to  impose  the  condition  that  in  the  parts  of  space  occu- 
pied by  the  fundamental  fluid  this  constant  shall  have  the  value 
zero.  This,  in  connection  with  the  general  condition  {B),  of  course 
involves  also  a  restriction  upon  the  generality  of  the  motion  of 
the  fluid  bodies.  The  nature  and  consequence  of  this  restriction 
will  be  discussed  later,  but  for  the  present  it  is  sufficient  for  us  to 
know  that  nothing  prevents  us  from  introducing  it. 
.  The  question  now  arises :  are  we  also  entitled  to  introduce  for 
the  hydrodynamic  system  a  condition  corresponding  to  the  condi- 
dition  (7)J  for  the  ether?  To  answer  this  we  must  refer  to  the 
dynamic  equation  (5,  c).  On  account  of  the  restriction  (Dj),  we 
shall  have  vot=  0  in  the  fundamental  fluid.  On  account  of  con- 
dition (Dg),  we  shall  have  div  A  =  0,  so  that  two  of  the  right  hand 
terms  of  the  equation  for  the  energetic  motion  disappear.  Writing 
a  J  =  a  4-  a^  and  remembering  the  condition  (^^3),  just  introduced, 
we  find  curl  a^  =  curl  a^,  and  the  equation  therefore  reduces  to 

da. 

Furthermore,  we  are  free  to  introduce  the  condition  that  the  ex- 
terior force  f  shall  be  zero  for  every  point  in  the  fundamental  fluid, 
so  that  the  equation  becomes 

"dt  ^  ~~  ^        ^«^  ^  ^' 
Now  if  at  any  point  in  space  a^  =  0,  we  shall  also  have 

^a       ^ 
dt      "' 

/.  e.,  under  the  given  conditions  there  can  be  no  energetic  field 
intensity  a^  unless  it  existed  previously.  The  same  will  be  true 
of  the  energetic  velocity  A^,  which  is  simply  proportional  to  the 
corresjwnding  field  intensity  a^.  Nothing  prevents  us,  conse- 
quently, from  requiring  that  in  the  space  occupied  by  the  funda- 
mental fluid  we  shall  have  the  condition 


PROPERTIES    OF   THE    IIYDKODYNAMIC    FIELD.  07 

(A)  A=0 

always  fulfilled.  For  evidently  we  have  the  right  to  introduce 
the  condition  (Z)J  as  an  initial  condition.  And,  as  we  have  seen, 
if  it  is  fulfilled  once,  it  will  always  be  fulfilled. 

Summing  up  the  contents  of  (/>>,)•  •  ■{T)^)  we  find  that  we  have 
introduced  the  following  conditions  defining  the  difference  between 
the  fluid  bodies  and  the  surrounding  fundamental  fluid,  which  is 
analagous  to  the  difference  between  the  bodies  and  the  surround- 
ing ether  in  the  electromagnetic  field.  The  fundamental  fluid  has 
constant  mobility  (specific  volume),  just  as  the  ether  has  constant 
inductivity  ;  the  fluid  bodies  may  have  a  mobility  varying  from 
point  to  point  and  differing  from  that  of  the  fundamental  fluid  ; 
just  as  the  bodies  in  the  magnetic  field  may  have  an  inductivity 
varying  from  point  to  point  and  differing  from  that  of  the  ether. 
The  fundamental  fluid  never  has  velocity  of  expansion  or  con- 
traction, E,  while  this  velocity  may  exist  in  the  fluid  bodies;  just 
as  in  the  free  ether  we  have  do  distribution  of  true  electrification 
or  magnetism,  while  such  distribution  may  exist  in  material  bodies. 
The  fundamental  fluid  never  has  a  distribution  of  dynamic  vortices, 
while  such  distributions  may  exist  in  the  fluid  bodies  ;  just  as  the 
ether  in  the  case  of  stationary  fields  never  has  a  distribution  of 
currents,  electric  or  magnetic,  while  such  distributions  may  exist 
in  material  bodies.  The  fundamental  fluid  never  has  an  energetic 
velocity,  while  this  velocity  may  exist  in  the  fluid  bodies ;  just  as 
the  ether  never  has  an  energetic  (impressed)  polarization,  while  such 
polarization  may  exist  in  material  bodies. 

Under  these  conditions  the  geometric  properties  of  the  hydro- 
dynamic  field  and  the  stationary  electric  or  magnetic  field  are  de- 
scribed by  equations  of  exactly  the  same  form.  Thus,  under  the 
given  conditions,  whose  physical  content  we  shall  consider  more 
closely  later,  there  exists  a  perfect  geometric  analogy  between  the 
two  kinds  of  fields. 

9.  Dynamic  Properties  of  the  Hydrodynamie  Field.  —  It  is 
easily  seen  that  under  certain  conditions  an  inverse  dynamic 
13 


98  FIELDS    OF    FORCE. 

analogy  will  be  joined  to  this  geometric  analogy.  For  let  us  im- 
pose the  condition  that 

shall  always  be  satisfied,  /.  e.,  that  the  energetic  specific  momentum 
shall  be  conserved  locally.  When  this  condition  is  fulfilled,  the 
equation  of  the  energetic  motion,  which  we  will  now  have  to  use 
for  the  bodies  only,  reduces  to 

{IS)  f  =  (div  A)a^  —  |a;^vot  +  (curl  aj  x  A, 

/.  e.,  if  the  condition  of  the  local  conservation  of  the  energetic 
specific  momentum  must  be  fulfilled,  there  must  act  upon  the  system 
an  exterior  force  f,  whose  distribution  per  unit  volume  is  given 
by  (b).  According  to  the  principle  of  equal  action  and  reaction, 
this  force  thus  balances  a  force  fj,  exerted  under  the  given  condi- 
tions by  the  fluid  system.  The  fluid  system  therefore  exerts  the 
force 
(^,)  fi  =  —  (div  A)a^^  +  ^a;v«  —  (curl  a  J  x  A, 

which,  in  form,  oppositely  corresponds  to  the  force  which  is  exerted, 
according  to  Heaviside's  investigation,  by  the  electric  or  the 
magnetic  field  in  the  corresponding  case. 

10.  Second  Form  of  the  Analogy.  —  The  physical  feature  of  the 
analogy  thus  found  is  determined  mainly  by  the  condition  (9,  a)  for 
the  local  conservation  of  the  energetic  specific  momentum.  The 
physical  content  of  this  condition  we  will  discuss  later.  But  first 
we  will  show  that  even  other  conditions  may  lead  to  an  analogy,  in 
which  we  do  not  arrive  at  Heaviside's,  but  at  some  one  of  the 
other  expressions  for  the  distribution  of  force. 

We  start  again  with  the  equation  of  motion, 

IdA 

Now,  instead  of  introducing  the  actual  specific  momentum  a^,  I 
introduce  at  once  the  induced  sj)ecific  momentum  a  and  the  ener- 
getic velocity  A,  according  to  the  equation  of  connection 

(^)  A  =  aa  +A  . 


PROPERTIES    OF    THE    HYDRODYNAMIC    FIELD. 


99 


Performing  the  differentiation  and  making  use  of  the  eciuation  of 
continuity  (2,  a),  we  have 

da.  1  dA 

-^  +  (divA)a+^^=f-Vy. 

Introducing  in  the  first  left  hand  term  the  U)cal  time-derivation, 
da.  1  ^^K 

or,  transforming  the  second  left  hand  member  according  to  the 
vector  formula, 

(c)        /^  +  Aa  V  +  (curl  a)  x  A  +  (^iv  A)a  +  ^  ^^  =  f  -  Vy>. 

Using  the  equation  of  connection  (6)  and  performing  simple  trans- 
formations, we  get  for  the  second  term  in  the  left  hand  member 

Aav  =  '^aav  +  A  av 
=  iava^  +  A^av 
=  V(j-^a-  +  A^  •  a)  -  |a- v«  -  aA  V- 

Introducing  this  in  {<■), 

da  .,     ,  .    .    _x   ,    1  ^^A 

(J) 

+  (curl  a)xA-aA^V  =  f-  V/>. 

Now,  we  can  split  the  equation  in  two,  requiring  that  the  vector 
a  satisfy  the  equation 

(e)  |^=_v(i.  +  ^^a^  +  A-a), 

and  we  find  that  the  other  vector  A  must  satisfy  the  equation 


^  +  V(iaa^  +  A -a)  +  ^  ^^^^+  (div  A)a-  ^aVot 


(/) 


^  ^  =  f  _  (div  A)a  +  H'va  -  (curl  a)  x  A  -h  aA^V. 


a    dt 


100  FIELDS    OF   FORCE. 

Both  equations  are  different  from  the  corresponding  equations  (5, 
6)  and  (5,  e).  But,  as  is  seen  at  once,  the  new  equation  for  the  in- 
duced motion  involves  the  same  geometric  property  as  the  previous 
one,  namely,  the  local  conservation  of  the  dynamic  vortex,  expressed 
by  (i>).  We  arrive  thus  at  the  same  set  of  fundamental  geomet- 
ric equations  as  before,  (A)  ■  ■  ■  (C).  Furthermore,  we  have  evi- 
dently the  same  right  as  before  to  introduce  the  restrictive  condi- 
tions (Dy),  {D.^,  {D^.  A  discussion  of  equation  {f),  similar  to 
that  given  above  for  equation  (5,  d),  shows  us  that  we  are  entitled 
in  this  case  also  to  impose  the  condition  (D^)  upon  the  fundamental 
fluid,  since  in  a  fluid  having  the  properties  (D,)  •  •  •  (Dg)  a  moving 
fluid  particle  cannot  have  an  energetic  velocity  if  this  did  not 
exist  previously. 

The  geometric  analogy  therefore  exists  exactly  as  before,  the 
conditions  for  its  existence  being  changed  only  with  respect  to  this 
one  point,  that  the  condition  (DJ  now  refers  to  the  material  parti- 
cles belonging  to  the  fundamental  fluid,  and  not  to  the  points  in 
space  occupied  by  this  fluid.  The  consequence  of  this  difference 
will  be  discussed  later. 

Finally,  we  see  that  to  this  geometric  analogy  we  can  add  a 
dynamic  analogy.  Requiring  that  the  energetic  velocity  be  con- 
served individuaUy,  we  have 

and,  reasoning  as  before,  we  find  that  under  this  condition  the  fluid 
system  will  exert  per  unit  volume  the  force 

(^2)        ^2=  —  (^iv  ^)^  +  ia-va  —  (curl  a)  x  A  +  aA^v, 

which,  in  form,  oppositely  corresponds  to  the  forces  in  the  electric 
or  magnetic  field,  according  to  the  expression  (IV.,  8  E^). 

11.  We  have  thus  arrived  in  two  different  ways  at  an  analogy 
between  the  e([uations  of  hydrodyuamic  fields  and  those  of  the 
stationary  electric  or  magnetic  field.  And,  from  an  analytical 
point  of  view,  this  analogy  seems  as  complete  as  possible,  apart 
from  the  ojijwsite  sign  of  the  forces  exerted  by  the  fields. 


PROPERTIES    OF    THE    IIYDRODYNAMIC    FIELD.  l^l 

In  regard  to  the  closeness  of  this  analytical  anology,  we  have  to 
remark   that  we  do  not  know  /vith   perfect  certainty  which  of 
the  expressions  {E^)  or  (E.^),  if  either,  represents  the  true  distribu- 
tion of  the  elementary  forces  in  the  electric  or  the  magnetic  field, 
while  the  corresponding  distribution  of  forces  in  the  hydrodynamic 
field  are  real  distributions  of  forces  which  are  exerted  by  the  field 
and  which  have  to  be  counteracted  by  exterior  forces,  if  the  condi- 
tions imposed  upon  the  motion  of  the  system  are  to  be  fulfilled.    A\  e 
cannot,  therefore,  decide  which  of  the  two  forms  that  we  have  found 
for  the  analogy  is  the  most  fundamental.     But  we  kno^  with  per- 
fect certainty  that,  if  we  integrate  this  system  of  elementary  forces 
for  a  whole  body,  we  get  the  true  value  of  the  resultant  force  in  the 
electric  or  magnetic  field.     When  we  limit  ourself  to  the  considera- 
tion of  the  resultant  force  only,  the  two  forms  of  the  analogy  are 
therefore  equivalent.     And  from  the  integration  performed  in  the 
preceding  lecture  we  conclude  at  once,  that  the  resultant  forces 
upon  the  bodies  in  the  hydrodynamic    field  can   also  be  repre- 
sented as  resulting  from  the  fictitious  distributions 

(^^)  fs  =  —  (div  A)a  -  A,av  -  (curl  a)  x  A, 

and 

(^j  f^  =  -  a^,  (div  a)a  -  a,  (curl  a)  x  a. 

The  fact,  which  we  have  just  i)roved,  that  the  laws  of  the  elec- 
tric or  magnetic  fields  and  of  the  hydrodynamic  fields  can  be  rep- 
resented by  the  same  set  of  formuke,  undoubtedly  shows  that  there 
is  a  close  relation  between  the  laws  of  hydrodynamics  and  the  laws 
of  electricity  and  magnetism.  But  the  formal  analogy  between  the 
laws  does  not  necessarilv  imply  also  a  real  analogy  between  the 
things  to  which  they  relate.  Or,  as  Maxwell  expressed  it:  the 
analogy  of  the  relations  of  things  does  not  necessarily  unply  an 
analogy  of  the  things  related. 

The  subiect  of  our  next  investigation  will  be,  to  consider  to 
what  extent  we  can  pass  from  this  formal  analogy  between  the 
hydrodynamic  formulae  and  the  electric  or  magnetic  formula^  to  an 
analogy  of  perfectly  concrete  nature,  such  as  that  represented  by 
our  experiments. 


YI. 

FURTHER  DEVELOPMENTS  AND  DISCUSSIONS  OF 
THE   ANALOGY. 

1.  According  to  the  systems  of  formulae  which  we  have  de- 
veloped, the  hydrodyiiamic  analogy  seems  to  extend  to  the  whole 
domain  of  stationary  electric,  or  stationary  magnetic  fields.  But 
according  to  our  elementary  and  experimental  investigation,  we 
arrived  at  two  different  analogies  which  were  wholly  detached 
from  each  other.  There  is  no  contradiction  involved  in  these  re- 
sults. In  our  analytical  investigation  we  have  hitherto  taken  only 
a  formal  point  of  view,  investigating  the  analogy  between  the  for- 
mal laws  of  hydrodynamics  and  of  electromagnetism.  If,  from  the 
analogy  between  the  formal  laws,  we  try  to  proceed  further  to  an 
analogy  between  the  different  physical  phenomena  obeying  them, 
we  shall  arrive  at  the  two  detached  fragments  of  the  analogy 
which  we  have  studied  experimentally. 

2.  Between  the  hydrodynamic  and  the  electric  or  magnetic 
systems  there  is  generally  this  important  difference.  The  hydro- 
dynamic  system  is  moving,  and  therefore  generally  changing  its 
configuration.  But  apparently,  at  least,  the  electric  or  magnetic 
systems  with  which  we  compare  them  are  at  rest.  The  corre- 
spondence developed  between  hydrodynamic  and  electromagnetic 
formulae  therefore  gives  only  a  momentary  analogy  between  the 
two  kinds  of  fields,  which  exist  under  different  conditions. 

To  get  an  analogy,  not  only  in  formulae  but  in  experiments, 
we  must  therefore  introduce  the  condition  that  the  bodies  in  the 
hydrodynamic  system  should  appear  stationary  in  space.  This 
can  be  done  in  two  ways.  First,  the  fluid  system  can  be  in  a 
steady  state  of  motion,  so  that  the  bodies  are  limited  by  sur- 
faces of  invariable  shapes  and  position  in  space.  Second,  the 
fluid  can  be  in  a  state  of  vibratory  motion,  so  that  the  bodies  per- 
form small  vibrations  about  invariable  mean  positions. 

102 


DEVELOPMENTS    AND    DISCUSSIONS    OF    THE    ANALOGY.       lOo 

3.  Steadij  State  of  Motion.  —  Tlie  first  form  of  the  analytical 
analogy,  in  which  we  supposed  local  conservation  of  the  energetic 
specific  momentum, 

(«)  -aj=o, 

immediately  leads  us  to  the  consideration  of  a  perfectly  steady 
state  of  motion,  at  which  we  arrive,  if  we  assume  besides  (a)  also 
the  local  conservation  of  the  induced  specific  momentum, 

which  is  perfectly  consistent  with  («).  But  in  the  case  of  a  steady 
state  of  motion  the  generality  of  the  field  is  very  limited,  on  ac- 
count of  the  condition  that  the  fluid,  both  outside  and  inside, 
moves  tangentially  to  the  stationary  surface  which  limits  the 
bodies. 

4.  Ir rotational  Circulation  Outside  the  Bodies.  —  As  the  motion 
outside  the  bodies  fulfills  the  condition  curl  a  =  0,  and,  in  conse- 
quence of  the  constancy  of  the  specific  volume,  a,,,  also  the  con- 
dition curl  A  =  0,  the  motion  in  the  exterior  space  will  be  the 
well  known  motion  of  irrotational  circulation,  which  is  only  possible 
if  the  space  be  multiply  connected.  If,- then,  there  is  to  be  any 
motion  of  the  exterior  fluid  at  all,  one  or  more  of  the  bodies  must 
be  pierced  by  channels  through  which  the  fluid  can  circulate. 
Bodies  which  have  no  channels  act  only  as  obstructions  in  the 
current,  which  exists  because  of  the  channels  through  the  other 
bodies.  The  velocity  or  the  specific  momentum  by  which  this 
motion  is  described  has  a  non-uniform  scalar  potential.  The 
stream-lines  are  all  closed  and  never  penetrate  into  the  interior 
of  the  bodies,  but  run  tangentially  to  the  surfaces.  The  corre- 
sponding electrodynamic  field,  with  closed  lines  of  force  running 
tangentially  to  the  bodies  and  having  a  non-uniform  potential, 
is  also  a  well  known  field. 

5.  Corresponding  Field  Inside  the  Bodies.  —  This  exterior  field 
can  correspond,  in  the  hydrodynamic,  as  well  as  in  the  electro- 


104  FIELDS    OF    FORCE. 

magnetic  case,  to  different  arrangements  in  the  interior  of  the 
bodies.  The  most  striking  restriction  on  the  exterior  field  is  the 
condition  that  the  lines  of  force  or  of  flow  shall  never  penetrate 
into  the  bodies.  In  the  magnetic  case  this  condition  will  always 
be  fulfilled  if  the  bodies  consist  of  an  infinitely  diamagnetic 
material,  and  a  field  with  these  properties  will  be  set  up  by  any 
distribution  of  electric  currents  in  these  infinitely  diamagnetic 
bodies.  The  hydrodynamic  condition  corresponding  to  zero  in- 
ductivity  is  zero  mobility.  The  bodies  then  retain  their  forms 
and  their  positions  in  space  as  a  consequence  of  an  infinite 
density  and  the  accompanying  infinite  inertia.  Now  in  the  case 
of  infinite  density  an  infinitely  small  velocity  will  correspond  to  a 
finite  specific  momentum.  We  can  then  have  in  these  infinitely 
heavy  bodies  any  finite  distribution  of  specific  momentum  and  of 
the  dynamic  vortex,  which  corresponds  to  the  electric  current,  and 
yet  to  this  specific  momentum  there  will  correspond  no  visible 
motion  which  can  interfere  with  the  condition  of  the  immobility 
of  the  bodies. 

Other  interior  arrangements  can  also  be  conceived  which, pro- 
duce the  same  exterior  field.  The  condition  of  infinite  diamag- 
netivity  may  be  replaced  by  the  condition  that  a  special  system 
of  electric  currents  be  introduced  to  make  bodies  appear  to  be 
infinitely  diamagnetic.  The  corresponding  hydrodynamic  case  will 
exist  if  we  abandon  the  infinite  inertia  as  the  cause  of  the  immo- 
bility of  the  bodies  and  also  dispense  with  the  creation  of  any  gen- 
eral distribution  of  dynamic  vortices  in  the  bodies,  and  if  we  in- 
troduce instead,  special  distributions  of  vortices,  subject  to  the 
condition  that  they  be  the  vortices  of  a  motion  which  does  not 
change  the  form  of  the  bodies  or  their  position  in  space.  This 
distribution  of  the  dynamic  vortices  will,  from  a  geometric  point 
of  view,  be  exactly  the  same  as  the  distribution  of  electric  current 
which  makes  bodies  appear  infinitely  diamagnetic. 

Finally,  a  third  arrangement  is  possible.  In  bodies  of  any  in- 
ductivity  we  can  set  up  any  distribution  of  electric  currents,  and 
simultaneously  introduce  a  special  intrinsic  magnetic  polarization 


DEVELOPMENTS   AND   DISCUSSIONS    OF   THE    ANALOGY.      lOo 

which  makes  the  bodies  appear  to  be  infinitely  diamagnetic.  Cor- 
respondingly, we  can  give  to  bodies  of  any  mobility  any  distribu- 
tion of  dynamic  vortices  under  the  condition  that  we  fix  the 
bodies  in  space  by  a  suitable  distribution  of  energetic  velocities 
produced  by  external  forces. 

6.  The  Dynamic  Analogy.  —  In  the  cases  thus  indicated  the 
geometric  analogy  between  the  fields  will  be  perfect.  And  with 
this  direct  geometric  analogy  we  have  an  inverse  dynamic  analogy. 
The  system  of  elementary  forces,  by  which  the  field  tends  to  pro- 
duce visible  motions  of  the  bodies,  and  which  must  be  counter- 
acted by  exterior  forces,  oppositely  corresponds  in  the  two  systems. 

The  simplest  experiments  demonstrating  these  theoretical  results 
are  those  showing  the  attraction  and  the  repulsion  of  rotating  cylin- 
ders, and  the  attraction  of  a  non-rotating,  by  a  rotating  cylinder, 
corresponding  to  the  repulsion  of  a  diamagnetic  body  by  an  elec- 
tric current. 

As  the  analogy  thus  developed  holds  for  any  arrangement  of 
electric  currents  in  infinitely  diamagnetic  bodies,  it  will  also  hold 
for  the  arrangement  by  which  magnets  can  be  represented  accord- 
ing to  Ampere's  theory.  We  can  thus  also  get  an  analogy  to 
magnetism,  but  in  a  peculiarly  restricted  way,  since  it  refers  only 
to  permanent  magnets  constructed  of  an  infinitely  diamagnetic 
material.  The  hydrodynamic  representation  of  a  magnet  is  there- 
fore a  body  pierced  by  a  multitude  of  channels  through  which  the 
exterior  fluid  circulates  irrotationally.  Such  bodies  will  then  exert 
apparent  actions  at  a  distance  upon  each  other,  corresponding  in- 
versely to  those  exerted  by  permanent  magnets  which  have  the 
peculiar  property  of  being  constructed  of  an  infinitely  diamagnetic 
material.  This  peculiar  analogy  was  discovered  by  Lord  Kelvin 
in  1870,  but  by  a  method  which  differs  completely  from  that 
which  we  have  followed  here. 

7.  Besfricted  Generality  of  the  Field  for  the  Case  of  Vibratory 
Motion.  —  The  hypothesis  of  a  vibratory  motion  also  restricts  the 
generality  of  the  field,  but  in  another  way  than  does  the  condition  of 
steady  motion.      For,  when  the  specific  momentum  is  vibratory,  its 

14 


106  FIELDS    OF   FORCE. 

curl,  if  it  has  any,  must  also  be  vibratory.  But  we  have  found  that 
this  curl,  or  the  dynamic  vortex  density,  is  a  constant  at  every 
point  in  space,  and  is  thus  independent  of  the  time.  The  dynamic 
vortex  therefore  must  be  everywhere  zero,  and  the  equations  ex- 
pressing the  geometric  analogy  reduce  to 

A  =  aa  +  A^, 

(a)  curl  a  =  0, 

div  A=  B, 

with  the  conditions  for  the  surrounding  fluid, 

(6)  a=a^,  JE=0,  A=0. 

The  equations  thus  take  the  form  of  the  equations  for  the  static 
electric,  or  the  static  magnetic  field,  so  that  the  analogy  will  not 
extend  beyond  the  limits  of  static  fields.  To  establish  the  cor- 
responding dynamic  analogy  we  may  use  neither  of  the  conditions 
(V.,  9,  a  or  10,  </).  For  both  are  contradictory  to  the  condition 
for  vibratory  motion.  We  have  to  return  to  the  unrestricted 
equation  for  the  energetic  motion,  and  the  form  which  in  this  case 
leads  to  the  most  general  results  is  (10,  f),  which  according  to  (a) 
reduces  to 

(c)  ^  '-^^  =  f  -  (div  A)a  +  la^Va  +  aA  V. 

This  system  of  equations  is  valid  for  any  single  moment  during  the 
vibratory  motion.  We  shall  have  to  try  to  deduce  from  it  another 
system  of  equations  which  represents  the  invariable  mean  state  of 
the  system. 

8.  Periodic  Inunctions.  —  To  describe  the  vibratory  motion  we 
shall  employ  only  one  periodic  function  of  the  time,  and  therefore 
the  different  particles  of  the  fluid  will  not  have  vibratory  motions 
independent  of  each  other.  The  motion  of  the  fluid  will  have 
the  character  of  a  fundamental  mode  of  an  elastic  system.  To 
describe  this  fundamental  mode  we  use  a  periodic  function,  y,  of  the 
period  t  ;  thus 

(«)  At  +  r)  =f{t). 


DEVELOPMENTS    AND   DISCUSSIONS    OF   THE    ANALOGY.      107 

The  values  of  the  function  /shoukl  be  contained  between  finite 
limits,  but  the  period  t  should  be  a  small  quantity  of  the  first 
order.  Further,  the  function  /  must  be  subject  to  the  following 
conditions  :  during  a  period  it  shall  have  a  linear  mean  value  0, 
and  a  quadratic  mean  value  1,  thus 

Evidently  these  conditions  do  not  restrict  the  nature  of  the  func- 
tion, provided  it  be  periodic.  Any  periodic  function  may  be  made 
to  fulfil  them  by  the  proper  adjustment  of  an  additive  constant  and 
of  a  constant  factor.  An  instance  of  a  function  which  fulfils  the 
conditions  is 

(d)  f{t)  =  V2,  sin  27r  (  ^  +  h  j  • 

From  the  conditions  that  the  period  is  a  small  quantity  of  the 
first  order  and  that  the  mean  linear  value  of  the  function  for 
a  period  is  zero,  it  is  deduced  at  once,  that  the  time  integral  of  the 
function  over  any  interval  of  time  multiplied  by  any  finite  factor 
n  will  never  exceed  a  certain  small  quantity  of  the  first  order.  AVe 
may  thus  write 

(e)  Jl'nf{t)dt<h, 

where  n  is  a  finite  factor,  and  h  a  certain  small  quantity  of  the 
first  order. 

9.  Representation  of  the  Vibratory  State  of  3fotion  by  Quantities 
Independent  of  the  Time.  —  To  get  equations  which  define  uni- 
formly the  vibratory  motion  we  can  now  make  use  of  the  property 
of  the  field,  that  it  is  determined  uniquely  by  the  energetic  veloc- 
ity A^  in  connection  with  the  velocity  of  expansion  E.  The  motion 
will  thus  be  definitely  determined  by  the  two  equations, 

(«.)  A  =  A,, Jit), 


108  FIELDS    OF    FORCE. 

where  A,„,  aud  E^^  are  quantities  independent  of  the  time,  but 
varying  of  course  from  particle  to  particle.  As  to  their  absolute 
values,  these  constants  are  the  quadratic  mean  values  of  the  ener- 
getic velocity  A^  and  of  the  velocity  of  expansion  E.  For,  from 
equation  (8,  c),  we  get 

1        /*t+T 

F'  =  -  i      Ehlt. 

The  constants  A,,„  and  £",„  for  different  particles  in  space  may 
have  different  signs.  These  are  always  given  by  the  equations 
(Oj)  and  (a^),  and  the  rule  of  signs  may  be  expressed  thus ;  the 
quantities  A,„,  and  E„,  have  respectively  always  the  same  sign  as 
the  variable  quantities  A  and  E  had  at  a  certain  initial  time.  The 
absolute  signs  thus  attributed  to  A,^  and  E„^  have  no  great  im- 
portance, but  it  is  important  that  this  rule  determines  perfectly  the 
signs  which  the  different  quantities  A,„,  and  E,,^  have  relatively  to 
each  other. 

With  regard  to  the  motions  determined  by  («),  we  can  conclude 
from  the  property  (8,  e)  of  the  function/,  that  the  energetic  velocity 
produces  displacements  from  the  mean  position  of  the  particle, 
which  never  exceed  a  certain  small  quantity  of  the  first  order. 
And  in  the  same  way  we  conclude,  that  the  change  of  volume  pro- 
duced by  the  periodic  velocity  of  expansion  and  contraction  never 
exceeds  a  small  quantity  of  the  same  order.  This  has  the  impor- 
tant consequence  that,  neglecting  small  quantities  of  the  first  order, 
we  can  consider  the  specific  volume,  oc^  of  the  fluid  as  constant,  ex- 
cept, of  course,  in  cases  where  it  has  to  undergo  a  differentiation 
with  respect  to  the  time. 

According  to  this,  it  is  easy  to  write  the  explicit  expressions  of 
the  actual  velocity  A  and  of  the  specific  momentum  a.     Doing 
this, 
{\)  A  =  A„,/(0, 


DEVELOPMENTS   AND   DISCUSSIONS    OF   THE    ANALOGY.      109 

For  the  substitution  of  these  expressions  and  the  expressions  (a) 
in  the  equations  (7,  a)  shows  that  they  satisfy  them,  if  the  quanti- 
ties independent  of  the  time  satisfy  the  equations 

(c)  curl  a„^  ^  0, 
div  A    =  E  , 

in  connection  with  the  conditions  for  the  exterior  fluid 

(d)  a  =  a^,  A,„,  =  0,  E„^  =  0. 

If  these  equations  be  satisfied,  (6)  will  satisfy  the  equations  and 
represent  the  solution,  as  there  exists  but  one. 

The  equations  (c),  which  the  quantities  A,„,  a„„  A,,„„  E,„,  satisfy, 
have  exactly  the  same  form  as  the  equations  (7,  a).  They  give, 
therefore,  for  all  times  the  same  analogy  to  an  invariable  electro- 
magnetic field  as  the  corresponding  variable  quantities  give  for  a 
single  moment.  The  similarity  is  so  great  that  it  is  not  even 
necessary  to  introduce  two  sets  of  notation.  To  pass  from  the 
one  form  of  the  analogy  to  the  other  it  is  sufficient  to  change  the 
signification  of  the  letters  in  the  equations  (7,  a) ;  if  these  quan- 
tities are  interpreted,  not  as  the  velocities  and  the  specific  mo- 
menta themselves,  but  as  representing  in  the  indicated  manner  the 
mean  intensities  of  these  quantities,  they  give  the  geometric 
analogy  existing  at  any  time  between  the  electric  or  magnetic 
field  and  the  case  of  vibratory  motion  in  the  hydrodynamic  field. 

1 0.  The  Mean  Value  of  the  Force  in  the  Vibratory  Field. —  Fi  nally , 
to  examine  the  dynamics  of  the  field  we  have  to  substitute  the 
expressions  (9,  a)  and  (9,  b)  in  the  equation  of  energetic  motion 
(7,  c)  and  perform  the  integration  over  a  period  of  the  oscilla- 
tions. Using  the  property  (8,  b)  of  the  function  /,  we  find  that 
the  left  hand  member  of  the  equation  disappears.  Designating  by 
f^l  the  mean  value  of  the  exterior  force  f  and  using  the  property 
(8,  e)  of  the  function/,  we  find 

0  =  f,;  -  (div  AJa,,  +  iaf. v«  +  a,„A.„, v. 


110  FIELDS    OF    FORCE. 

This  equation  shows  that  during  the  vibratory  state  of  motion 
the  external  force  will  have  to  balance  a  mean  force  exerted  by 
the  system,  which  has  the  value 

f    =  —  (div  A  )a    +  |a-  v^  +  a,„A,,„,  v. 

The  expression  has  again  exactly  the  same  form  as  the  expres- 
sion for  the  force  in  the  case  of  the  momentary  analogy,  except  that 
the  varying  quantities  are  replaced  by  quantities  independent  of 
the  time.  The  similarity  of  the  expressions  makes  it  unnecessary 
to  use  two  systems  of  notation.  We  can  write  the  expression  for 
the  force 

f^  =  —  (div  A)a  +  |a"V«  +  aA^V, 

and  interpret,  according  to  the  circumstances,  the  quantities  a  and 
A  as  the  momentary  values  of  specific  momentum  and  velocity,  re- 
spectively or  as  the  quantities  which  represent  in  the  way  indicated 
the  mean  intensities  of  these  quantities.  In  one  case  we  arrive  at 
the  analogy  which  exists  for  a  moment  only,  in  the  other  case  at 
the  analogy  which  exists  independent  of  the  time.  Both  analogies 
have  the  same  degree  of  exactness,  the  geometric  analogy  being 
direct,  and  the  dynamic  analogy  being  inverse. 

11.  We  have  thus  arrived  at  this  result,  that  in  the  case  of 
vibratory  motion  the  hydrodynamic  field  can  be  described  with 
reference  to  geometric  properties  by  the  following  formulae, 

A  =  aa  +  A^, 
(a)  curl  a  =  0, 

div  A  =  B, 

together  with  the  conditions  for  the  fundamental  fluid, 

(6)  ^=%,         B=0,         A  =  0. 

And  this  fluid  system,  in  the  supposed  vibratory  state  of  motion, 
will  exert  exterior  forces  tending  to  produce  visible  motions,  which 
are  given  by 
(c)  f ^  =  —  (div  A)a  +  |a'-vst  +  aA^v. 


DEVELOPMENTS   AND    DISCUSSIONS    OF   THE    ANALOGY.      Ill 

In  these  equations  all  quantities  arc  independent  of  the  time. 

But  these  equations  are  also  the  fundamental  equations  for  an 
electrostatic  or  for  a  magnetic  system,  except  for  the  diiference  that 
the  force  f^  has  the  opposite  direction.  It  is  an  open  question 
whether  this  expression  for  the  elementary  forces  in  the  case  of  the 
electric  or  magnetic  field  is  fundamental,  or  only  a  fictitious  force 
which  gives  the  right  value  of  the  resultant  force  upon  the  whole 
body. 

We  have  succeeded  in  proving  this :  the  vibratory  hydrodynamic 
field  has  the  same  geometric  configuration  as  an  electrostatic  or  a 
magnetic  field.  In  tJie  hydrodynamic  field  there  are  forces  whose 
resultant  upon  finite  bodies  oppositely  corresponds  to  the  correspond- 
ing resultant  forces  in  the  electric  or  magnetic  field. 

To  show  that  this  result  gives  the  full  explanation  of  all  our 
experiments  with  the  pulsating  and  oscillating  bodies  we  have 
only  to  add  one  remark.  In  our  experiments  we  used  pulsating 
and  oscillatinsr  bodies  constructed  of  solid  material.  On  the 
other  hand,  in  our  mathematical  developments  we  have  consid- 
ered the  bodies  as  fluid.  But  these  fluid  bodies  are  subject  to  the 
action  of  forces  which  give  the  prescribed  state  of  vibration,  and 
which  are  subject  to  no  restrictive  conditions.  Nothing  prevents 
us,  therefore,  from  adjusting  these  forces  so  as  to  give  the  fluid 
bodies  the  same  motion  as  they  would  have  if  they  were  con- 
structed of  solid  material.  The  reactions  exerted  upon  them  by 
the  surrounding  fluid  will  then  of  course  be  exactly  the  same  as 
if  they  were  constructed  of  solid  material. 

12.  We  have  nothing  to  add  to  the  demonstration  of  the  anal- 
ogy. But,  to  make  ourselves  better  acquainted  with  it,  we  may 
make  a  simple  application  of  it.  In  the  analogy,  for  instance, 
pulsating  particles  produce  fields  of  the  same  geometric  configura- 
tion as  electrically  charged  particles,  and  are  acted  upon  by  forces 
oppositely  corresponding  to  those  acting  upon  the  latter.  Pulsating 
particles  will  therefore  act  upon  each  other  according  to  a  law 
analogous  to  that  of  Coulomb,  except  for  the  reversed  sign  of  the 
force.  Introducing  for  the  charges,  or  the  intensities  of  pulsation 
of  the  two  particles 


112  FIELDS    OF   FORCE. 

e  =  fEch,         e  =  jE'dr, 
and  using  the  rational  system  of  units,  we  get  for  this  law 

a^  47rr" 

r  being  the  distance  between  the  two  particles,  and  a^  the  induc- 
tivity,  or  the  mobility  of  the  medium. 

Let  us  now  imagine  an  investigator  who  observes  the  attraction 
and  the  repulsion  of  the  pulsating  bodies,  but  who  is  not  capable 
of  observing  the  water  which  transfers  the  action,  or  the  pulsa- 
tions which  set  up  the  field  in  the  water.  He  will  then  believe 
that  he  sees  an  action  at  a  distance,  following  a  law  having  the 
same  form  as  that  governing  the  action  at  a  distance  between  elec- 
trified particles. 

Let  us  imagine  that,  as  he  proceeds  in  his  further  investigations, 
he  moves  one  pulsating  body,  e',  from  point  to  point  in  the  space 
surrounding  the  other,  measures  at  each  point  the  force  F,  and 
draws  an  arrow  representing  the  value  of  F/e'.  He  then  arri.ves 
at  the  formal  disposition  of  a  field  which  is  associated  with  the 
pulsating  body  e.  He  has,  no  more  than  in  the  electrical  case, 
a  formal  right  to  attribute  to  this  field  a  physical  significance,  or 
to  attribute  to  the  recorded  vector  a  physical  existence.  His  ex- 
periments give  him  evidence  only  of  this,  that  there  is  a  force  act- 
ing at  the  point  where  he  places  his  second  charge,  e'.  But  he 
has  no  evidence  of  the  existence  of  a  physical  vector  at  this  point 
after  he  has  removed  the  charge  e'. 

But  in  spite  of  this,  he  may  try  to  change  his  view.  He  may 
imagine  the  existence  of  a  medium  which  he  does  not  see,  and 
make  the  hyj)othcsis  that  the  vector  represents  some  state  exist- 
ing, or  some  process  going  on,  in  this  medium.  In  the  electrical 
case  we  have  no  direct  evidence  that  this  hypothesis  is  correct, 
although  thus  far,  the  development  of  our  knowledge  of  electricity 
makes  it  extremely  probable  that  there  must  be  some  truth  in  it. 
But  in   the  hydrodynamic  case  we  have  the  full  evidence:  the 


DEVELOPMENTS    AND    DISCUSSIONS    OF    THE    ANALOGY.      113 

medium  exists  ;  it  is  an  incompressible  fluid.  And  the  vector  re- 
corded represents  tlie  specific  momentum  in  the  field  set  up  in  the 
fluid  by  the  pulsating  body.  Thus  we  get  a  verification  by  analogy 
of  the  hypothesis  which  forms  the  basis  of  the  whole  modern 
theory  of  electricity. 

13.  But  now  let  our  hydrodynamic  investigator  proceed  still 
further.  Let  him  conclude  with  Maxwell,  that  the  attraction 
and  repulsion  between  the  pulsating  bodies  must  depend  upon 
a  stress  in  the  medium.  Following  Maxwell's  developments 
he  will  arrive  at  the  expression  of  Maxwell's  stresses,  with 
the  reversed  sign.  But  his  conclusion  in  this  case,  that  Max- 
w- ell's  stresses  exist  in  the  fluid  and  produce  the  attraction  of  the 
pulsating  body,  is  wrong.  The  stress  that  exists  in  the  fluid  and 
produces  the  apparent  actions  at  a  distance  is  not  Maxavell's 
stress,  but  the  isotropic  stress  or  pressure  in  the  fluid.  We  can- 
not conclude  from  this  that  Maxwell's  developments  are  also 
wrong  for  the  electric  field.  But  we  have  full  evidence  that  they 
may  be  wrong,  even  in  this  case. 

To  return  to  the  hydrodynamic  case,  it  is  easy  to  point  out 
where  the  error  comes  in.  Maxwell  only  introduces  his  stresses 
to  account  for  the  forces  which  produce  the  visible  motions.  But 
in  the  hydrodynamic  field  the  stress  or  pressure  has  a  double 
task;  first,  to  maintain  the  field,  and  second,  to  produce  the  visible 
motions.  And  it  is  extremely  remarkable  that  the  stress  which 
has  this  double  eifect  is  a  stress  of  much  simpler  nature  than  the 
stress  imagined  by  Maxwell,  which  produces  only  one  of  the 
two  effects. 

When  we  developed  the  electromagnetic  equation  of  activity 
according  to  Heaviside,  we  also  met  with  the  more  general 
stresses  introduced  by  him,  which  reduce  in  simple  cases  to 
Maxwell's  stresses.  We  cannot  test  Heaviside's  develop- 
ments in  the  same  way  as  Max\\-ell's.  For  we  have  no 
hydrodynamic  analogy  extending  to  the  electromagnetic  phe- 
nomena of  the  most  general  type,  from  which  he  starts  when  he 
forms  the  equation  of  activity.  But  the  fact  remains  that  the 
15 


114  FIELDS    OB"    FORCE. 

srtesses,  even  in  Heaviside's  theory,  are  introduced  only  to  explain 
the  visible,  motion  observed  in  the  field,  not  the  formation  or  main- 
tenance of  the  field  itself.  And  even  Heaviside  gives,  while  em- 
phasizing the  im])ortance  of  the  stress-problem,  in  different  forms 
expression  to  the  unsatisfactory  nature  of  our  present  solution  of 
it.     Thus 

'^  Our  attitude  towards  the  general  application  of  the  special 
form  of  the  stress  theory  obtained  should,  therefore,  be  one  of 
scientific  scepticism.  This  should,  however,  be  carefully  distin- 
guished from  an  obstinate  prejudice  founded  upon  ignorance,  such 
as  displayed  by  some  anti-Maxwellians,  •  •  •  * 

"  It  is  natural  to  ask  Avhat  part  do  the  stresses  play  in  the  prop- 
agation of  disturbances  ?  The  stresses  and  accompanying  strains 
in  an  elastic  body  are  materially  concerned  in  the  transmission  of 
motion  through  them,  and  it  might  be  thought  that  it  might  be 
the  same  here.  But  it  does  not  appeur  to  be  so  from  the  electro- 
magnetic equations  and  their  dynamical  consequences  —  that  is  to 
say,  we  represent  the  propagation  of  disturbances  by  particular 
relations  between  the  space-  and  the  time-variations  of  E  and  H ; 
and  the  electromagnetic  stress  and  possible  motions  seem  to  be 
accompaniments  rather  than  the  main  theme."t 

It  may,  therefore,  be  a  question  whether  this  will  not  be  the 
great  problem  in  the  theory  of  electricity,  to  find  a  stress  which 
accounts  for  both  the  formation  and  propagation  of  the  electro- 
magnetic field  and  for  the  visible  motions  of  the  charged  or  polar- 
ized bodies,  just  as  the  pressure  in  the  fluid  accounts  for  both  the 
formation  of  the  hydrodynamic  field  and  for  the  visible  motions 
of  the  pulsating  or  oscillating  bodies. 

*  Electromagnetic  Theory,  Vol.  I,  p.  87. 
t  Loc.  cit.,  p.  110. 


VII. 

GENERAL   CONCLUSIONS. 

Remarks   on   Methods  of  Research  and  of  Instruction 
IN  Theoretical  Physics. 
1.    The   Problem  of  Fields   of  Force.— ^Ng  have  iu  the  pre- 
ceding lectures  taken  the  terra  "field  of  force"  in  a  more  general 
sense  than  usual.     From  the  electric  or  magnetic  fields  we  have 
extended  this  term  also  to  the  fields  of  motion  in  a  perfect  fluid. 
And  this    has  been  perfectly  justified  by   the  results  obtained, 
the  most  striking  of  which  is  the  extraordinary  analogy  in  the 
properties  of  the  two  kinds  of  fields.     So  fiir  as  the  analogy   ex- 
tends, there  is  one,  and  only  one,  diiference,  the  reversed  sign 
of  the  energetic  forces.     The  relation  of  the  electromagnetic  and 
the  hydrodynamic   fields    may  be  compared  to  the  relation   be- 
tween an  object  and  its  image  in   a  mirror  ;  every  characteristic 
detail  of  the  object  is  recognized  in   the  image,  but  at  the  same 
time   there  is  the   characteristic    difference   that    left  and    right 
are  interchanged.     But,  however  peculiar  this  diiference  may  be, 
it  cannot  hide  the  common  structure  of  the  object  and  its  image. 
The  discovery  of  this  extraordinary  analogy  gives  rise  to  sev- 
eral considerations,  and  one  of  the  first  is  this  :  Has  our  research 
been  exhaustive?     Are  the  phenomena   investigated    by  us   the 
only  phenomena  which  have  the  same  general   structure  as  the 
electromagnetic  phenomena,  or  can  still  other  phenomena  with 
corresponding  fundamental  properties  be  discovered  ?         ^ 

I  think  that  it  is  very  improbable  that  our  investigation  has 
been  exhaustive.  Even  within  the  domain  of  hydrodynamics  our 
investigation  has  probably  been  incomplete.  There  are,  indeed, 
very  strong  indications  that  an  analogy  between  electromagnetic 
and  hydrodynamic  fields  may  be  found  with  quite  another  cor- 
respondence between  the  electric  and  the  hydrodynamic  quanti- 

115 


116  FIELDS    OF    FOKCE. 

ties.  And  if  we  no  longer  limit  ourselves  to  the  consideration  of 
fluids,  but  pass  to  media  of  other  and  more  general  properties,  we 
may  hope  to  find  still  other  forms  of  the  analogy,  perhaps  of  even 
greater  generality. 

2.  Fields  in  Other  Media  than  Fluids. — The  question  now 
arises:  Are  not  the  laws  which  we  have  found  so  entirely  depen- 
dent upon  the  fluid  properties  that  it  will  be  useless  to  look  for 
similar  laws  when  we  pass  to  other  media?  To  answer  this  ques- 
tion we  have  to  look  for  the  origin  of  the  hydrodynamic  analogy. 
We  then  see  that  the  geometric  analogy  had  its  origin  to  a  great 
extent  in  the  equation  of  continuity.  And,  as  this  equation  ex- 
presses the  principle  of  the  conservation  of  mass,  it  holds  for  any 
material  medium,  and  furnishes  the  same  basis  for  a  possible  geo- 
metric analogy  to  electromagnetic  fields. 

On  the  other  hand,  the  dynamic  properties  of  the  hydrodynamic 
fields  had  their  origin  principally  in  the  inertia  of  the  fluid  masses. 
This  is  seen  equally  well  in  the  elementary  development  of  the 
forces  by  the  principle  of  kinetic  buoyancy  and  in  the  mathematical 
developments  of  Lecture  Y,  where  it  is  seen  that  the  complete 
expression  of  the  energetic  force  develops  from  the  inertia  term  of 
the  hydrodynamic  equation  of  motion. 

A  brief  consideration  thus  shows  that  the  principal  conditions 
from  which  the  hydrodynamic  analogy  to  the  electromagnetic  fields 
developed,  exist  in  any  material  medium,  not  alone  in  fluids. 
But  the  special  form  which  the  analogy  will  take,  its  accuracy,  and 
its  extent,  will  depend  upon  the  special  properties  of  the  different 
media.  Thus  the  special  properties  of  fluids  admitted  the  exist- 
ence of  an  analogy  whicli  is  perfectly  accurate,  if  we  except  the 
inverse  nature  of  the  forces,  but  limited  in  extent.  It  will  there- 
fore be  a  most  fascinating  subject  for  research  to  examine  whether 
there  exist  media  in  which  the  accuracy  of  the  analogy  is  pre- 
served, while  its  scope  is  widened.  Or,  in  other  words,  to  deter- 
mine the  dynamic  conditions  of  a  medium  in  which  the  analogy, 
with  unaltered  precision,  has  the  greatest  possible  extent. 

3.  The  Fields  in  a  Transverse  Elastic  Medium. — To  examine 


GENERAL    CONCLUSIONS.  117 

the  chances  of  progress  along  this  line  it  will  bo  advantageous  to 
consider  briefly  the  fields  in  a  medium  with  the  common  elastic 
solid  properties.  Now  it  is  well  known  that  there  is  an  exten- 
sive geometric  analogy  between  the  fields  of  motion  in  an  elastic 
medium  with  properly  adjusted  constants  and  the  electromagnetic 
fields  of  the  most  general  type.  The  coexistence  and  equivalence 
of  the  two  theories  of  light,  the  elastic  and  the  electromagnetic, 
proves  this  perfectly.  Indeed,  the  electromagnetic  theory  of  light 
originated  from  the  analogy  which  Maxwell  succeeded  in  stating 
between  the  equations  for  optical  phenomena,  developed  by  FiiES- 
NEL  and  his  successors  from  the  hypothesis  of  the  transverse  elas- 
tic ether,  and  the  equations  which  he  had  himself  developed  to 
describe  electromagnetic  fields. 

We  will  consider  this  analogy  in  the  simplest  possible  case.  Let 
the  medium  be  homogeneous  and  isotropic,  and,  furthermore,  in- 
compressible and  subject  to  the  action  of  no  exterior  force.  U  being 
the  vector  displacement,  a  the  specific  volume,  and  fi  the  constant 
of  transverse  elasticity,  the  equation  of  motion  of  the  nicdiimi  is 

generally  written 

1    d^V 

(«)  -^  =  /.v-u. 

As  a  and  fi  are  constants,  this  may  be  written 

^-^  =  vVU. 

On  the  right  hand  side  of  the  equation  we  can  now  introduce 
the  velocity 

W  ^  =  -et' 

This  member  may  at  the  same  time  be  written  in  a  modified  fi)rni, 
the  operation  V"^  being,  for  the  solenoidal  vector  U,  equivalent  to 
—  curF.      The  equation  may  then  be  written 

(c)  —  =  —  curl-  a/LtU  =  —  curl  aV  I  curl      U   j. 


118  FIELDS    OF    FORCE. 

Let  us  introduce  now 

,  1 
{d)  B  =  -  curl  -  U, 

from  which  we  get 

dB  1   d\J  ,1 

^rr  =  —  curl ^  =  —  curl  -  A, 

dt  a    at  a    ' 

or,  if  we   introduce  the  specific   momentum   a  according    to  the 
equation 

(e)  A  =  aa, 

we  have 

^B 


On  the  other  hand,  the  introduction  of  {(l)  in  (c)  gives 

dA 


{g)  ^  =  curl  fia'B. 


If  we  introduce 

b  =  /^a-B, 
((j)  finally  takes  the  form   . 

(A)  -^  =  cind  b. 

Thus  we  can  substitute  for  equation  (a)  the  following  system  of 

equations 

dA 

-,-  =  curl  b, 
ot 

clB 

-—  =  —  curl  a, 

dt 

where  the  vectors  A  and  a,  B  and  b  are  connected  by  the  equa- 
tions 

A  =  aa, 

B  =  /Sb, 
where  /S  has  the  signification 


GENERAL    CONCLUSIONS.  119 

But  this  system  is  the  system  of  Maxwell's  equations  for  a 
medium  which  is  electrically  and  magnetically  homogeneous  and 
isotropic,  and  which  is  the  seat  of  no  intrinsic  electromotive  or 
magnetomotive  forces.      And  we  get  the  following  correspondence : 

A  electric  flux velocity 

a  electric  field  intensity  ..  .specific  momentum 

B  magnetic  flux. curl  of  specific  mass-displacement 

b  magnetic  field  intensity  . .  (curl  of  sp.  mass-displacement)  tin?' 

a  electric  inductivity specific  volume 

/S  magnetic  inductivity  .  .  .  .  density^ /coeff.  of  elasticity 

As  is  well  known,  we  are  free  to  give  different  forms  to  this 
geometric  analogy.  We  have  used  this  freedom  to  choose  a  form 
which  makes  the  analogy  a  direct  continuation  of  the  hydrody- 
namic  analogy. 

The  extent  of  this  geometric  analogy  is  very  great  even  though 
we  have  avoided  full  generality  by  neglecting  heterogeneities  and 
intrinsic  forces.  For  it  extends  now  to  that  point  where  the  cross- 
ing of  electric  and  magnetic  phenomena  takes  place,  the  point  at 
which  the  hydrodynamic  analogy  ceased. 

4.  Dynamics  of  the  Field  in  the  Transverse  Elastic  Medium. — 
These  well  known  developments,  which  lead  to  the  geometric 
analogy  of  electromagnetic  and  elastic  fields,  apparently  give  not 
the  faintest  indication  of  the  existence  also  of  a  dynamic  analogy, 
corresponding  to  that  which  we  know  from  the  investigation  of 
the  hydrodynamic  field,  which  is  quite  the  opposite  of  what  we 
should  expect  from  our  preceding  considerations. 

The  explanation  of  this  apparent  contradiction  is,  however,  im- 
mediate. As  we  have  remarked,  the  energetic  force  in  the  hydro- 
dynamic  fields  originated  in  the  inertia  term  of  the  hydrody- 
namic equation.  But  the  equation  of  motion  of  the  elastic  medium, 
as  it  is  generally  written  (3,  a),  contains  this  term  incompletely, 
the  local  time  derivation  djdt  being  used  as  a  first  approxima- 
tion for  the  individual  derivation  djdt,  which  would  give  to  the 
left  member  of  the  equation  its  proper  form. 


120  FIELDS    OF   FORCE. 

Let  us  repeat,  therefore,  the  preceding  development,  but  start- 
ing with  the  equation 

(«)  ^  ^^-  =  /.v-u  +  f, 

in  which  the  left  member  has  its  exact  form,  and  in  which  we  have 
added  on  the  right  hand  side  the  exterior  force  f,  which  we  sup- 
pose, however,  small  in  comparison  to  the  elastic  forces.  The  left 
hand  member  of  this  equation  is  identical  with  the  left  hand  member 
of  the  hydrodynamic  equation,  and  may  be  developed  in  exactly 
the  same  way.     We  may  thus  write,  as  in  (V,  10), 

A  =  aa  -f  A^, 
and  then  equation  («)  in  the  form 

da.  1  f/A 

-^j  +  V(Aa!a-  +  A    a)  +  ^  ^^  +  (div  A)a  -  laVa, 

+  (curl  a)  X  A  —  aA^v  =  ftV'U  +  f, 

corresponding  to  (V,  10,  d).  As  the  medium  is  supposed  homo- 
genous and  incompressible,  this  equation  reduces  to 

<5a  ,  ,        1  c?A        ,      ,     , 

^^  +  V  (laa^  +  A  •  a)  -h  -  ^f  -h  (curl  a)  x  A 

—  aA^V  =  f^v'^U  +  f. 

This  may  now  be  introduced  in  equation  («),  and  the  equation 
then  separated  into  two  equations,  just  as  in  the  case  of  the  cor- 
responding hydrodynamic  equation.  We  thus  arrive  at  the  sys- 
tem of  equations 

t'^a 

(h)  ^^  =  /^v'U  -  v(aA  -f  i^a-), 

1  clA 

(c)  ^  'df^^'^  ^^'^  ~  ^"""'^  ^^  ^  ^' 

where  the  first  is  that  of  tlie  "  induced,"  the  second  that  of  the 
"  energetic  "  motion. 

The  first  of  these  equations  differs  from  equation  («)  only  by 


GENERAL    CONCLUSIONS.  121 

quantities  of  the  order  generally  neglected  in  the  theory  of  elas- 
ticity. If  we  agree  to  neglect  these  quantities,  we  may  still  de- 
scribe the  geometry  of  the  field  by  the  system  of  equations 

dA 

~-  =  curl  b, 

dB 


where  now 


--r  =  —  curl  a, 
ot 


A  =  eta  4-  A^, 
B=  /3b. 


But  if  we  proceed  to  the  second  approximation,  we  have,  besides 
these  equations  describing  the  geometric  configuration  from  time 
to  time,  to  consider  another  partial  motion,  given  by  equation  (c). 
And  if  we  demand  here  that  the  energetic  velocity  be  conserved 
individually,  dAJdt  =  0,  we  find  that  an  exterior  force  f  must 
be  applied,  which  has  the  value 

f  =  aA^v  —  (curl  a)  x  A. 

This  force  inversely  corresponds  to  the  exterior  force  which  had  to 
be  applied  in  the  corrfesponding  electromagnetic  system,  in  order 
to  prevent  the  production  of  visible  motions  due  to  the  forces 
exerted  by  the  system  upon  intrinsic  electric  polarization,  cor- 
responding to  A,  and  upon  magnetic  current,  —  curl  a. 

5.  This  result  thus  gives  a  new  and  remarkable  extension  of 
the  analogy.  And  the  fact  that  continued  research  leads  to  further 
extension  of  the  analogy  between  the  formal  laws  of  the  phenom- 
ena, if  not  between  the  phenomena  themselves,  seems  to  indicate 
that  there  exists  a  common  set  of  laws,  the  laws  of  the  fields  of 
force,  where  the  expression  fields  of  force  is  taken  in  a  suitably 
extended  sense.  If  this  be  true,  the  investigation  of  this  common 
set  of  laws  and  the  discovery  of  all  i)henomena  obeying  them  will 
be  one  of  the  great  problems  of  theoretical  physics.  And  investi- 
16 


122  FIELDS    OF   FORCE. 

gations  suggested  by  this  idea  may  perhaps,  sooner  or  later,  lead 
even  to  the  discovery  of  the  true  nature  of  the  electric  or  mag- 
netic fields. 

6.  But  investigations  of  this  kind  can  be  considered  as  only 
just  begun.  And  if  we  return  to  our  result  relating  to  the 
elastic  field,  it  is  easy  to  point  out  its  incompleteness.  In  this 
field  we  have  not  only  the  well  known  geometric  analogy,  but 
also  a  dynamic  analogy  to  the  electrodynamic  field,  at  least  so 
long  as  we  confine  our  attention  to  the  analogy  between  the 
formal  laws  of  the  phenomena,  and  not  to  the  phenomena  them- 
selves. And  this  dynamic  analogy  has  exactly  the  same  inverse 
nature  as  in  the  case  of  the  hydrodynamic  field.  But  it  should 
be  emphasized  that  this  dynamic  analogy,  in  the  form  in  which 
we  have  found  it,  has  not  the  same  degree  of  completeness  as 
the  geometric  analogy.  I  pass  over  here  the  fact  that  we  have 
given  to  our  development  only  a  restricted  form,  by  supposing 
the  medium  to  be  homogeneous  and  incompressible,  and  thus  ex- 
cluding beforehand  heterogeneities  and  changes  of  volume.  Most 
likely  this  gap  can  be  filled.  But  the  great  drawback  is  this  :  the 
dynamics  of  the  electromagnetic  field  relates  to  two  classes  of  forces, 
the  electric  forces  and  the  magnetic  forces,  while  our  analysis  of 
the  elastic  field  has  led  us  to  the  discovery  of  only  one  class  of 
forces,  namely,  forces  which  correspond  to  the  electric  forces,  ac- 
cording to  our  interpretation  of  them ;  but  we  have  discovered 
no  trace  of  forces  corresponding  to  the  magnetic  forces  of  the  elec- 
tromagnetic field.  It  is  true  that,  making  use  of  the  symmetry, 
we  can  change  the  interpretation,  comparing  from  the  beginning 
the  velocity  with  the  magnetic,  instead  of  the  electric  flux.  The 
elastic  field  will  then,  according  to  our  analysis,  give  forces  cor- 
responding to  the  magnetic  forces  of  the  electromagnetic  field,  but 
at  the  cost  of  the  complete  disappearance  of  the  forces  which  pre- 
viously corresponded  to  the  electric  forces. 

7.  Final  Remarks  on  the  Problem  of  Fields  of  Force.  —  It  is  too 
early  of  course  to  consider  tliis  incompleteness  as  a  decisive  failure 
of  the  analogy  in  the  elastic  media.     From  the  beginning  there 


GENERAL    CONCLUSIONS.  123 

seemed  to  exist  no  dynamic  analogy  at  all.  However,  writing  the 
inertia-term  of  the  elastic  equation  in  its  correct  form,  we  found 
at  once  forces  corresponding  to  one  class  of  forces  in  the  elec- 
tromagnetic field.  But  even  in  this  form,  the  elastic  equations 
will  generally  be  only  approximations.  For  the  expression  of  the 
elastic  forces  is  based  upon  Hooke's  law  of  the  proportionality  of 
the  stresses  to  the  deformations,  and  this  law  is  an  approxima- 
tion only.  Will  the  addition  of  the  neglected  terms,  under  cer- 
tain conditions,  bring  full  harmony  between  the  electromagnetic 
and  the  elastic  field  ?  I  put  this  question  only  to  emphasize  a 
problem  which  is  certainly  worth  attention.  If  the  research  be 
carried  out,  it  will  certainly  lead  to  valuable  results,  whether  the 
answer  turns  out  to  be  positive  or  negative.  And  even  if  the 
answer  be  negative,  the  investigation  of  the  fields  of  force  will 
not  therefore  be  completed.  It  is  not  at  all  to  be  expected  that 
the  intrinsic  dynamics  of  the  electromagnetic  field  should  corre- 
spond to  that  of  one  of  the  simple  media  of  which  we  have  a 
direct  empirical  knowledge.  When  the  fields  of  these  simple 
media  are  thoroughly  explored,  so  that  we  know  how  far  the 
analogy  of  their  fields  to  those  of  electromagnetism  goes,  the  time 
will  then  have  come,  I  think,  to  put  the  prol)lem  in  another  form  : 
What  should  be  the  properties  of  a  medium,  whose  fields  shall  give 
the  completest  possible  analogy  to  electromagnetic  fields  ? 

Even  when  the  problem  is  put  in  this  form,  we  have  the  advan- 
tage that  preparatory  work  of  great  value  has  already  been  done. 
The  gyrostatic  ether,  which  was  introduced  by  MacCullagh  and 
Lord  Kelvin,  is  a  medium  with  very  remarkable  properties.  As 
is  well  known,  the  fields  in  this  medium  give  as  perfect  a  geo- 
metric analogy  to  the  electromagnetic  field  as  the  elastic  medium. 
And  the  form  of  the  expression  for  the  energy  in  this  medium 
seems  to  indicate  the  possibility  of  a  dynamic  analogy  of  greater 
extent  than  that  which  is  likely  to  be  found  in  the  case  of  the 
common  elastic  medium. 

It  will  be  clear  after  these  few  remarks,  that  the  problem  of  fields 
of  force  is  of  vast  extent.      We  are  only  at  the  beginning  of  it. 


124  FIELDS    OF    FORCE. 

8.  Kinetic  Theories.  —  The  problem  of  fields  of  force  iu  this 
general  sense  evidently  belongs  to  a  class  of  problems  which  has 
been  present  in  the  minds  of  the  natnral  philosophers  from  the 
very  beginning  of  our  speculations  with  regard  to  nature  ;  but 
the  method  of  stating  the  problem  has  changed. 

From  the  very  first  of  human  speculations  on  the  phenomena 
of  nature  strong  efforts  have  been  made  to  construct  dynamic 
models  of  these  phenomena.  These  dynamic  models  seem  to  be 
the  natural  way  to  render  the  phenomena  of  nature  intelligible 
to  the  human  mind.  I  need  only  remind  you  of  the  efforts 
of  the  old  philosophers  of  the  atomistic  school,  such  as  Demok- 
RiTOS  or  Epicurus,  or  of  philosophers  of  later  time,  like  Des- 
cartes. Or  I  may  mention  a  long  series  of  theories  of  special 
physical  phenomena,  for  instance  Huyghen's,  and  Newton's 
theories  of  light,  theories  opposed  to  each  other,  but  both  of  them 
dynamic  theories.  Or  I  may  remind  you  of  the  kinetic  theory  of 
gases  of  Bernoulli,  Kronig,  Clausius,  and  Maxwell,  or  of 
Maxwell's  ingenious  ideas  of  "  physical  lines  of  force." 

But  most  of  these  speculations  have  broken  down  more  or  less 
completely.  Of  the  universal  constructions  of  the  atomists  nothing 
is  left  except  the  building  stones  themselves,  the  atoms,  whichj 
however,  have  remained  to  this  day  an  indispensable  idea  to  the 
natural  philosopher.  Descartes'  theory  of  universal  vortices 
had  the  same  fate.  But  though  it  fell,  it  left  germs  of  fruitful 
ideas,  leading  in  the  direction  of  the  fields  of  force.  Newton's 
theory  of  light  also  broke  down.  But  it  did  not  exist  in  vain. 
For  the  fact  that  phenomena  of  radiation  could  be  explained  ac- 
cording to  his  principle  immensely  facilitated  the  interpretation  of 
the  new  phenomena  of  radiation,  discovered  in  vacuum  tubes  and 
in  radioactive  substances.  The  theory  of  light  of  Huyghens  and 
Fresnel  is  still  unshaken,  if  it  is  considered  merely  as  an  abstract 
undulation  theory.  But  it  is  open  to  doubt  whether  it  still  exists 
in  its  original  form  as  a  theory  which  explains  the  phenomena  of 
light  on  dynamic  principles.  For  a  dynamic  theory  of  light  will 
hardly  be  satisfactory  before  we  have  a  dynamic  theory  of  electro- 
magnetism. 


GENERAL   CONCLUSIONS.  125 

This  fate  of  dynamic  theories  which  have  had  the  unanimous 
support  of  all  physicists  may  also  bring  into  a  dubious  light  dy- 
namic theories  which  are  still  highly  appreciated,  as,  for  instance, 
the  kinetic  theory  of  gases.  As  a  matter  of  fact,  a  strong  reaction 
against  dynamic  theories  has  appeared. 

9.  The  Relation  of  Kinetic  Theories  to  the  Phenomenological 
Principles  of  Research.  —  Reactions  against  exaggerations  are 
always  wholesome.  On  the  other  hand,  it  is  a  law  of  nature  that 
reactions  usually  go  to  exaggerations.  In  accordance  with  this 
law,  the  energetic  school  developed.  I  will  not  enter  upon  the 
exaggerations  of  this  school.  But  it  has  done  good  by  em- 
phasizing phenomenological  research,  the  principles  of  w'hich 
were  developed  especially  by  Professor  Mach  at  Vienna,  pre- 
vious to  the  formation  of  the  energetic  school,  and  without  its 
exaggerations. 

The  leading  principle  of  Professor  Mach  is,  that  the  phenomena 
of  nature  should  be  investigated  with  perfect  impartiality  and  free- 
dom from  prejudice  that  the  research  should  lead  ultimately  to  a 
kinetic  theory,  or  to  any  other  preconceived  view  of  natural  phe- 
nomena. If  this  idea  be  carried  out  with  perfect  consistency,  it  is 
necessary,  of  course,  not  only  that  we  should  avoid  the  positive 
prejudice  that  the  physical  phenomena  are  ultimately  phenomena 
of  pure  kinetics,  but  that  we  should  also  avoid  the  negative  preju- 
dice that  the  phenomena  of  nature  are  not  ultimately  kinetic. 
The  principles  of  phenomenological  research  are  therefore,  rightly 
understood,  not  hostile  to  kinetic  research,  if  this  be  only  con- 
ducted with  perfect  impartiality. 

If  this  be  admitted,  the  extreme  importance  of  kinetic  research 
will  not  be  denied  by  the  adherents  of  the  phenomenological 
principles  of  research.  For  no  unprejudiced  observer  will  deny 
that  physical  phenomena  are  inextricably  interwoven  with  kinetic 
phenomena.  Neither  will  he  deny  that  our  power  of  kinetic 
research  exceeds  by  far  our  power  of  every  other  kind  of  physical 
research.  The  reason  is  obvious.  We  are  all  kinetic  machines. 
Instinctive  kinetic  knowledge  is  laid  down  in  our  muscles  and 


126  FIELDS    OF    FORCE. 

nerves  as  an  inheritance  from  the  accumulated  dynamic  exper- 
ience of  our  ancestors,  and  has  been  further  developed  without 
interruption  from  the  time  of  our  first  motions  in  the  cradle.  And 
furthermore,  while  we  have  this  invaluable  instinctive  knowledge 
of  the  fundamental  principles  of  dynamics,  we  have  at  the  same 
time  an  objective  view  of  dynamic  phenomena  as  of  no  other 
physical  phenomena,  from  the  fact  that  we  have  the  power  of 
following  and  controlling  the  phenomena  of  motion  by  several  of 
our  senses  at  the  same  time,  while  for  other  phenomena,  such  as 
sound,  light,  or  heat,  we  have  only  one  special  sense,  and  for  still 
others  such  as  electricity,  magnetism,  or  radioactivity,  we  have 
no  special  senses  at  all. 

No  wonder,  therefore,  that  at  the  time  when  science  grew  np 
dynamics  soon  developed  into  the  model  science,  from  the  formal 
point  of  view  the  most  perfect  of  physical  sciences,  and  in  this 
respect  second  only  to  pure  mathematics.  This  also  explains  why 
the  dynamic  side  of  physical  phenomena  has  always  offered  the 
best  point  of  attack  for  research,  while,  on  the  other  hand,  it  gives 
the  obvious  reason  why  we  may  be  tempted  to  overestimate  the 
value  of  our  dynamic  constructions. 

But  if  a  reaction  against  exaggeration  has  been  necessary,  noth- 
ing can  be  gained  by  giving  up  advantages  which,  for  subjec- 
tive reasons  at  least,  are  combined  with  the  kinetic  direction  of 
research,  whatever  be  the  final  objective  result  of  these  researches. 
The  reaction  has  taught  us  that  problems  should  be  stated  in  a 
perfectly  unprejudiced  way. 

10.  The.  Coiiipnrative  Method.  —  It  is  such  a  way  of  conducting 
the  investigation  of  the  relations  betw^een  physics  and  kinetics, 
which  we  have  tried  to  realize  in  these  researches  on  fields  of  force. 
The  essence  of  the  method  is,  that  kinetic  systems  are  made  the 
subject  of  pure  phenomenological  research.  Their  laws  and  pro- 
perties are  made  the  subject  of  impartial  investigation,  but  w'ith 
constant  attention  to  the  analogies  and  the  contrasts  between  the 
laws  found  for  the  dynamic  system  and  the  laws  of  physical 
phenomena. 


GENERAL   CONCLUSIONS.  127 

And  this  comparative  method  is  applicable  far  outside  the 
limits  of  our  special  problem  of  fields  of  force.  Indeed,  it  is  the 
method  used  by  such  authors  as  Boltzmaxx,  Helmholtz, 
Hertz,  and  Wir.LARD  Gibbs,  in  their  profound  researches  in  the 
dynamical  illustration  of  physical  laws  and  phenomena,  especially 
those  of  heat  and  thermodynamics.  These  researches  are  un- 
completed, just  as  are  our  researches  on  fields  of  force,  and  will 
probably  remain  so  for  a  long  time.  But  the  more  they  have 
advanced,  the  stronger  has  been  the  demand  for  rigorousness  of 
methods  ;  the  more  have  the  methods  of  construction  been  forced 
back  and  the  impartial  comparative  method  advanced.  And  no 
one  has  emphasized  conservative  and  safe  methods  more  strongly 
than  WiLLARD  Gibbs.  In  the  preface  to  the  last  work  which  he 
has  left  us  he  expresses  this  in  the  following  plain  words  : 

"Difficulties  of  this  kind  have  deterred  the  author  from  at- 
tempting to  explain  the  mysteries  of  nature,  and  have  forced  him 
to  be  contented  with  the  more  modest  aim  of  deducing  some  of 
the  more  obvious  propositions  relating  to  the  statistical  branch  of 
mechanics.  Here  there  can  be  no  mistake  in  regard  to  the  agree- 
ment of  the  hypotheses  with  the  facts  of  nature,  for  nothing  is 
assumed  in  that  respect.  The  only  error  into  which  one  can  fall, 
is  the  want  of  agreement  between  the  premises  and  the  conclu- 
sions, and  this,  with  care,  one  may  hope,  in  the  main,  to  avoid." 
His  method  is  exactly  the  same  as  that  which  we  have  tried  to 
■  employ,  namely  the  impartial  research  of  each  branch  of  physics 
by  itself,  but  with  comparison  of  the  resulting  laws,  and  with  the 
greatest  possible  caution  with  respect  to  the  conclusions  to  be 
drawn  from  the  analogies  and  the  contrasts  presenting  themselves. 
The  method  is  that  of  comparative  anatomy.  Is  it  too  sanguine 
a  hope,  that  this  method  will,  sooner  or  later,  unveil  for  us  the 
relations  of  the  different  physical  phenomena,  just  as  the  methods 
of  comparative  anatomy  successively  give  us  an  insight  into  the 
relation  between  the  different  kinds  of  living  beings? 

11.   On  the  Value  of  the   Comparative  Method  for  Instruction 
in  Theoretical  Physics.  — I  cannot  leave  the  discussion  of  this  com- 


128  FIELDS    OF    FORCE. 

parative  method  without  seizing  the  occasion  to  emphasize  its  vahie 
also  in  instruction  in  theoretical  physics.  The  results  obtained 
by  this  method  and  the  discovery  of  similar  laws  in  apparently 
perfectly  different  branches  of  physics  makes  an  unexpected  con- 
centration of  instruction  possible.  And  if  the  principle  be  carried 
out,  and  similar  facts  presented  in  similar  ways,  the  analogies  will 
facilitate,  to  a  degree  not  to  be  overestimated,  the  power  of  the 
student  to  comprehend  and  assimilate  the  matter.  Especially  will 
this  be  the  case  when  the  analogies  give  us  the  opportunity  to 
throw  light  upon  obscure  theories,  such  as  those  of  the  electromag- 
netic field,  by  means  of  perfectly  plain  and  comprehensible  theories 
such  as  those  of  the  hydrodynamic  field,  in  which  every  step  can 
be  made  by  rigorous  mathematical  conclusions,  by  elementary  in- 
ductive reasoning,  or  by  experiment. 

And  yet,  this  saving  of  labor,  so  imperatively  demanded  in  our 
days  whenever  possible,  is  perhaps  less  essential  in  comparison 
with  the  independence  relative  to  the  methods  and  the  results, 
which  the  student  will  gain  when  he  observes  how  similar  methods 
can  be  used,  and  similar  laws  obtained,  in  apparently  widely  dif- 
ferent branches  of  physics.  This  will  teach  him  to  judge  better  the 
value  of  the  methods,  and  give  him  independence  of  view  for  his 
future  work  as  an  investigator. 

The  arrangement  of  instruction  according  to  principles  by  which 
the  analogies  at  our  disposal  are  used  as  much  as  possible  for  the 
benefit  of  the  student,  is  a  problem  which  has  its  own  charm,  in- 
voluntarily attracting  the  attention  of  the  investigator  engaged  in 
research  on  these  analogies.  Time  does  not  allow  me  to  enter 
upon  the  details  of  my  experiments  in  this  direction.  But  before 
concluding  these  lectures,  I  wish  to  answer  an  objection,  which 
seems  to  lie  near  at  hand,  against  the  use  to  a  greater  extent  of 
these  analog-ies  in  instruction. 

12.  Theory  and  Practice.  —  It  seems  to  be  an  obvious  reflec- 
tion, that  instruction  conducted  according  to  the  plan  thus  indi- 
cated will  be  of  an  exceedingly  abstract  nature,  tending  to  develop 
in  a  purely  theoretical  direction,  and  to  draw  attention  away  from 


GENERAL    CONCLUSIONS. 


129 


practically  useful  points.  To  take  the  nearest  example:  hydro- 
dynamics is  useful  if  it  teaches  us  to  understand  and  calculate 
water  motions  occurring  practically.  Now  water  is  practically 
homogeneous  and  incompressible,  and  hydrodynamics  of  practical 
use  will  have  to  direct  the  attention  to  the  investigation  of  the 
motions  of  this  simple  medium,  and  not  to  the  abstract  fluid  sys- 
tems considered  by  us,  with  density  and  compressibility  varying 
according  to  laws  never  occurring  practically. 

I  was  of  this  opinion  myself  when  I  commenced  my  study  of 
these  extraordinary  fluid  systems.     Nothing  was  further  from  my 
thoughts  than  to  expect  practical  results  from   investigations  of 
this  abstract  nature.     But  as  the  result  of  conversations  with  sci- 
entific friends  who  were  interested  in  the  dynamics  of  the  ocean 
and  the  atmosphere,  I  happened    to   see   that   certain   theorems 
which  I  had  developed  to  investigate  the  motions  of  my  abstract 
fluid  system  had  immediate  bearing  upon  the  motion  of  these  two 
media."     And  the  reason  why  these  theorems  had  not  been  discov- 
ered a  long  time  before  was  obvious.     To  work  out  the  science 
of   the  motion  of   fluids   in   a  practical   form    investigators    had 
always  considered  the  fluids  as  homogeneous  and  incompressible, 
or,  in   the   most  general  case,  as  compressible  according  to  an 
idealized  law,  so  that  the  density  depended   upon  the  pressure 
only.     But  these  very  suppositions  precluded  from  consideration 
the  primary  causes  of  the  motions  in  the  atmosphere  and  the 
sea.     For  these  primary  causes  are  just  the  difi^erences  of  density 
which  do  iwt  depend  upon  the  pressure,  but  on  other  causes,  such 
as  diff-erences  of  temperature  and  salinity  in  the  sea,  and  diifer- 
ences  of  temperature  and  humidity  in  the   atmosphere.     While 
the  old  theorems  of  the  practical  hydrodynamics  did  not  allow 
us  to  take  up  from   the   beginning  the  discussions  of  the  circu- 
lations of  the  atmosphere  and  the  sea,  the  thoerems  which  I  had 
developed  for  my  impractical  fluid  systems  gave  at  once  a  very 
simple  view  of  the  atmospheric  and  oceanic  circulations.    If,  there- 
fore, it  be  considered  a  question  of  practical  importance   to  mas- 
ter   the    dynamics  of   these    two  universal  media  on   which   we 
17 


130  FJEl.DS    OF    FORCE. 

human  beings  are  in  such  a  state  of  dependence,  then  the  methods 
of  this  theoretical  hydrodynamics  are  not  impractical.  And  I 
do  not  think  that  this  is  an  isolated  fact.  For  the  more  we  ad- 
vance in  theoretical  and  practical  research,  the  more  we  shall  dis- 
cover, I  think,  that  there  is  really  no  opposition  between  theory 
and  practice. 

I  hope  that  you  will  allow  me  to  exemplify  this  in  the  addi- 
tional lecture  to-morrow,  in  which  I  shall  consider  the  hydrody- 
namic  fields  of  force  in  the  atmosphere  and  the  sea. 


APPENDIX. 

Vector  Notation  and  Vector  Formulae. 

A  vector  with  the  rectangular  components  A^,  A^,  A^  is  desig- 
nated by  A. 

A  vector  with  the  rectangular  components  B^,  B^,  B^  is  desig- 
nated by  B. 

A  vector  with  the  rectangular  components  C^,  C ,  C,  is  desig- 
nated by  C. 

Vector  Sum.  —  The  three  scalar  equations, 

A   +  B  =C, 

X       '  X  X' 

A   +B  =C, 

are  represented  by  one  vector  equation, 

(1)  A  +  B  =  C. 

C  is  called  the  vector  sum  of  the  two  vectors  A  and  B, 

Scidat-  Product.  —  The  scalar  quantity  A^B^  -j-  A^B^  +  AB,  is 
designated  by  AB  and  called  the  scalar  or  dot-product  of  the 
vectors  A  and  B, 

(2)  kB  =  AB,+AB^  +  AB,, 

Vector  Product.  —  The  three  scalar  equations, 

AB,^-AB^^=C^, 
AB^-A^B  =  C,^, 
AB^-AB^=C,^, 

are  represented  by  one  vector  equation, 

(3)  A  X  B  =  C. 

The  vector  C  is  called  the  vector-  or  cross-product  of  the  two 
vectors  A  and  B.     The  definition  states  that  the  vector  product 

131 


132  FIELDS    OF    FOllCE. 

C  is  normal  to  each  of  the  vector-factors  A  and  B,  and  is  directed 
so  that  the  positive  rotation  according  to  the  positive  screw  rule 
around  the  vector  C  rotates  the  first  vector- factor,  A,  towards  the 
second,  B.  Change  of  the  order  of  the  factors,  therefore,  changes 
the  sign  of  the  vector-product. 

Triple  Products.  —  In  a  scalar  product  one  vector-factor  can  be 
a  vector-product.  For  this  triple  product  it  is  easily  proved  that 
dot  and  cross  can  be  interclianged,  and  that  circular  permutation 
of  the  factors  is  allowable,  thus 

.^.  ABxC  =  CAxB  =  BCxA 

^  ^  =AxBC  =  CxAB  =  BxCA. 

In  a  vector-product  one  factor  itself  may  be  a  vector-product. 
Cartesian  development  easily  gives  the  formula 

(5)  A  X  (B  X  C)  = -(AB)C  + (AC)B. 

Linear  Derivdtion  of  a  Scalar  Qaanfiti/.  —  The  three  scalar 
equations, 

da 
A  =  ;^j 

da 

A=^, 

oz 

are  represented  by  one  vector  equation, 

(6)  A  =  \70L. 

The  differentiating  symbol  V  or  "  del  "  represents  a  vector  opera- 
tion with  the  three  component-operations  d jdx,  <->  jdy,  d jdz.  The 
vector  A  or  v^  sliows  the  direction  of  greatest  increase  of  the 
values  of  the  scalar  function  a,  and  represents  numerically  the 
i-ate  of  this  increase.  The  vector  —  vx  is  called  the  gradient  of 
the  scalar  quantity  a  (compare  the  classical  expressions  pressure- 
gradient,  temperature  gradient,  etc.). 


APPENDIX.  133 

Sjiher'wal  Derivation  of  a  Scalar  Quantity.  —  The  sum  of  the 
secoud  derivations  of  a  scalar  quantity  may  be  called  the  spheri- 
cal derivative  of  this  quantity,  and  the  operation  of  spherical  de- 
rivation may  be  designated  by  V^,  thus 

d'^a       d^'a       d^-a.  ., 

Divergence.  —  The  scalar  quantity  dAjdx  +  dAJdy  -\-  dAJdz 

is  called  the  divergence  of  the  vector  A,  and  designated  by  div 

A,  thus 

dA       dA^      dA^       ,.     ^ 

Curl.  —  The  three  scalar  equations, 

dy         dz   ~     '' 

dA^      dA^  _ 

'^dz  ~'dx  ~     y' 

dx  dy  * ' 

define  a  vector  C,  which  is  called  the  curl  of  the  vector  A,  and 
the  three  scalar  equations  are  represented  by  the  one  vector  equa- 
tion, 
(9)  curl  A  =  C. 

Spherical  Derivation  of  a  Vector.  —  The  three  scalar  equations, 

^'A        d^-A        d^A^  ^^, 


d'A^       d'A^       d^A 
'M'  "^  "a/"  ^  ^dz 


a- A  0"A,  ^ 


134  FIELDS    OF    FORCE. 

define  the  vector  C,  which  is  called  the  spherical  derivative  of  A, 
and  the  three  scalar  equations  are  represented  by  one  vector 
equation, 

(10)  V'A  =  C. 

Linear  Operations.  —  The  three  equations, 

,  dB,  dB  ,  dB,       ^ 

ox         •'  ^y  ^^ 

,   dB         ,   dB  ,  dB, 

^'  dx     +      "   dy    +      ~   dz 

,    dB         ,   dB  ,  dB        ^ 

^•^^  +  ^^aF  +  ^^^='^^' 

may  be  represented  by  one  vector  equation, 

(11)  AvB  =  C. 
The  three  scalar  equations, 

.   dB\.         ,   dB^        ,  dB        ^  , 

A — -  +  A  -    -^  +  A  ~--^=  CJ, 
"  dx    ^     '  dx  '  dx  " ' 


dB  ,  dB^        ,  dB 

dy  ■'   dy  dy 


,  dB,         ,   dB^        .  dB,       ^, 

^  dz  ''  dz  ~  dz  '^ 

may  be  represented  by  one  vector  equation, 

(12)  ABv=C'. 

Between  the  two   vectors  defined  by  (11)  and  (12)  there  is  the 

relation 

(l;i)  A  vB  =  AB  V  +  (curl  B)  X  A. 

t^pecial  Fornvihc  of  Tran.fonnidion.  —  Tlie   following  formulie 
are  easily  verified  by  cartesian  expansion  : 

(14)  div  aA  =  a  div  A  +  A-  V  '^j 


APPENDIX.  135 

(15)  div  (A  X  B)  =  —  A  •  curl  B  +  B  •  curl  A, 

(IB)  curl  (a  V  /3)  =  V  ^  X  V  /3. 

If  the  openition  curl  be  used  twice  in  succession,  Me  get 

(17)  curP  A  =  V  div  A  -  v'  A. 

Integral  Fonnuke.  —  If  dr  be  the  element  of  a  closed  curve 
and  (Is  the  element  of  a  surface  bordered  by  this  curve,  we  have 

(18)  jAr^r  = /curl  A-r/s 

(Theorem  of  Stokes).  If  ds  be  the  element  of  a  closed  surface, 
whose  normal  is  directed  positively  outwards,  and  dr  an  element 
of  the  volume  limited  by  it,  we  have 

(19)  f^'^^^  =  Jdiv  Af/r. 

Transformation  of  Integrak  Involving  Products.  —  Integrating 
the  formula  (16)  over  a  surface  and  using  (18),  we  get 

(20)  faS7l3-dr  =  /  V  a  x  V  /S  •  ds. 

Integrating  (14)  and  (15)  throughout  a  volume  and  using  (19), 
we  get 

(21)  J  A  •  V  Mh  =  —  fa  div  Adr  +  JaA  ■  ds, 

(22)  J  A  •  curl  Bdr  =  Jb  ■  curl  Ar/r  -  /a  x  B  ■  ds. 

If  in  the  first  of  these  integrals  either  a  or  A,  in  the  second  either 
A  or  B,  is  zero  at  the  limiting  surface,  the  surface  integrals  will 
disappear.  When  the  volume  integrals  are  extended  over  the 
whole  space,  it  is  always  supposed  that  the  vectors  converge  towards 
zero  at  infinity  at  a  rate  rapidly  enough  to  make  the  integral  over 
the  surface  at  infinity  disappear. 

Performing  an  integration  by  parts  within  a  certain  volume  of 
each  cartesian  component  of  the  expressions  (11)  and  (12)  and 
supposing  that  one  of  the  vectors,  and  therefore  also  the  surface- 
integral  containing  it,  disappears  at  the  bounding  surface  of  the 
volume,  we  find,  in  vector  notation. 


136  FIEI>DS    OF    FORCE. 

(23)  Ja  V  Bdr  =  -  /B  cliv  Adr, 

(24)  Jab  V  (h  =  -/BA  V  dr.  ■ 
Integrating  equation  (13)  and  making  use  of  (23),  we  get 

(25)  JB  div  A  dr  =  -J'AB  V  d-  r  /(curl  B)  x  A  dr. 

For  further  details  concerning  vector  analysis,  see  :  Gibbs-Wil- 
son,  Vector  Analysis,  New  York,  1902,  and  Oliver  Heaviside, 
Electromagnetic  Theory,  London,  1893. 


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